Extreme eigenvalues and eigenvectors for finite rank additive deformations of non-hermitian sparse random matrices

EXTREME EIGENV ALUES AND EIGENVECTORS F OR FINITE RANK ADDITIVE DEF ORMA TIONS OF NON-HERMITIAN SP ARSE RANDOM MA TRICES W ALID HA CHEM a , MICHAIL LOUV ARIS b , JAMAL NAJIM a a CNRS, LIGM (UMR 8049), Universit ´ e Gustave Eiﬀel, ESIEE Paris, F r anc e. Emails: walid.hachem@univ-eiﬀel.fr, jamal.najim@univ-eiﬀel.fr b Dep artment of Mathematics, Y ale University, New Haven, USA. Email: michail.louvaris@yale.e du Abstract. Consider a n ˆ n sparse non-Hermitian random matrix X n deﬁned as the Hadamard pro duct between a random matrix with centered independent and identically distributed en tries and a sparse Bernoulli matrix with success probabilit y K n { n where K n ď n (and possibly K n ! n ) and K n Ñ 8 as n Ñ 8 . Let E n be a deterministic n ˆ n ﬁnite-rank matrix. W e prove that the outlier eigen v alues of Y n “ X n ` E n asymptotically match those of E n . In the sp ecial case of a rank-one deformation, assuming further that the sparsity parameter satisﬁes K n " log 9 n and that the entries of the random matrix are sub-Gaussian, we describ e the limiting behavior of the pro jection of the righ t eigenv ector asso ciated with the leading eigen v alue on to the righ t eigenv ector of the rank- one deformation. In particular, we prov e that the pro jection behav es as in the Hermitian case. T o that end, we rely on the recent universalit y results of Brailo vsk aya and v an Handel [ BvH24 ] relating the singular v alue spectra of deformations of X n to Gaussian analogues of these matrices. Our analysis builds up on a recent framew ork introduced by Bordena v e et.al. 2022 [ BCGZ22 ], and amoun ts to showing the asymptotic equiv alence b etw een the reverse characteristic p olynomial of the random matrix and a random analytic function on the unit disc with explicit dependence on the ﬁnite-rank deformation. 1. Introduction and main resul ts The study of eigen v alue outliers in random matrix theory has a rich and w ell-established history , particularly in the symmetric and Hermitian settings, where additiv e ﬁnite-rank deformation often lead to predictable and well- understo o d sp ectral deviations. A landmark result by Baik, Ben Arous, and P ´ ech ´ e (BBP) demonstrated that for sample co v ariance matrices with Gaussian entries, ﬁnite-rank deformations can induce outlier eigen v alues that separate from the bulk spectrum once a critical threshold is exceeded; see [ BBAP05 ]. This so-called BBP transition was so on extended to general en tries b y Baik and Silverstein [ BS06 ] and has since b ecome a foundational concept in the ﬁeld, with extensions to more general settings such as co v ariance-type matrices [ P au07 ], Wigner-t ype matrices [ CCF09 ], and other deformed matrices [ BGN11 ]. Key to ols in these developmen ts include the resolv ent metho d, master equations, and momen ts of large p ow er. The non-symmetric / non-Hermitian setting in tro duces additional challenges; nevertheless, signiﬁcant progress has b een achiev ed. In particular, [ T ao13 ] and [ BC16 ] provide a complete characterization of the outlier distri- bution in the i.i.d. case, assuming ﬁnite fourth momen ts for the entries. More recently , the sparse circular la w has b een established under minimal moment assumptions in [ R T19 ] and [ SSS25 ]. Building up on these adv ances, we pro v e that outlier results con tin ue to hold across all sparsit y regimes. Our main technical to ol is the analysis of the rev erse c haracteristic p olynomial, as dev eloped in [ BCGZ22 ]. F urthermore, under additional assumptions on the matrix and its sparsit y parameter, w e establish a result concerning the right-eigen vector asso ciated with the largest eigen v alue in the case where the additiv e deformation has rank one. T o this end, w e compare sp ectral quantities of the matrix with those of an analogous Gaussian ensemble, leveraging univ ersalit y results from [ BvH24 ]. 1 2 EXTREME EIGENV ALUES AND EIGENVECTORS By also relying on the technique of [ BCGZ22 ], the author of the recen t pap er [ Han25 ] also deals with the outliers induced by ﬁnite rank deformations of square matrices with independent and identically distributed en tries. This pap er deals among others with the sparse Bernoulli case with a ﬁnite rank additive deformation, a mo del close to ours. The sparsity parameter of the Bernoulli elements is assumed to conv erge to inﬁnity at the rate n o p 1 q . In this situation, it is moreov er assumed in [ Han25 ] that the ﬁnite rank deformation has a ﬁnite n um b er of non-zero elements. These assumptions are not required in our pap er, where we only need the deformation to hav e a b ounded op erator norm. Moreo v er, we do not put any assumption on the rate of increase of the sparsity parameter. In addition, when our deformation is of rank one, w e also study the angle b etw een the eigenv ector associated to the outlier and the “true” vector, a problem not considered in [ Han25 ]. On the other hand, [ Han25 ] tackles the problem of the extreme eigenv alues of ﬁnite pro duct of matrices. Random additively deformed non-Hermitian matrices app ear in man y applied ﬁelds, such as natural and artiﬁcial neural net works where the random matrix Y n at hand represen ts the random interactions b etw een the neurons [ SCS88 , WT13 ]. W e may also cite theoretical ecology where Y n , which is often sparse, mo dels the in teractions among living species within an ecosystem [ Bun17 , ABC ` 24 ], see also the references therein. In these ﬁelds, the eigenv alue of Y n with the largest mo dulus pla ys a central role in describing the time ev olution of the activit y of n interacting neurons or of the abundances of the n sp ecies that constitute the ecosystem. W e in tro duce some notation b efore stating our results. 1.1. Notations. Let C ` “ t z P C : ℑ z ą 0 u . The cardinalit y of a set S , counting multiplicities, is denoted b y | S | . F or m P N , set r m s “ H if m “ 0 and r m s “ t 1 , . . . , m u otherwise. Let z P C and A, B Ă C , then d p z , A q “ inf ξ P A | z ´ ξ | and the Hausdorﬀ distance b etw een A ab d B , denoted by d H p A, B q is deﬁned by d H p A, B q “ max " sup z P A d p z , B q ; sup z P B d p z , A q * . When m ą 0, we denote as S m the symmetric group o v er the set r m s . Let } ¨ } b e the matrix op erator norm or the v ector Euclidean norm. F or a matrix M , denote by M ‹ its conjugate transp ose; if u, v are column vectors with equal dimension, then x u, v y “ u ‹ v . Denote b y I m the m ˆ m identit y matrix, or simply I if the dimension can b e inferred from the context. Denote by σ p M q “ t λ 1 p M q , . . . , λ m p M qu the sp ectrum of a m ˆ m matrix M , b y ρ p M q its spectral radius, and b y s m p M q its least singular v alue. F or a m ˆ m matrix M “ p M ij q m i,j “ 1 and I , J Ă r m s , let M I , J “ p M ij q i P I ,j P J and M I “ p M ij q i,j P I . Denote by adj p M q the adjugate of M , i.e. , the transp ose of M ’s cofactors matrix. F or a vector x P C m and I Ă r m s let x I “ p x i q i P I . F or a sequence of random v ariables p U n q and a random v ariable U with v alues in a common metric space, denote by U n P Ý Ñ n Ñ8 U and U n law Ý Ñ n Ñ8 U the conv ergence in probabilit y and in law, resp ectiv ely . Let U n and V n b e random v ariables in some metric space with probability distribution µ n and ν n . The notation U n „ V n p n Ñ 8q refers to the fact that the sequences p µ n q and p ν n q are relativ ely compact, and that ż f dµ n ´ ż f dν n Ý Ý Ý Ñ n Ñ8 0 ´ ô E f p U n q ´ E f p V n q Ý Ý Ý Ñ n Ñ8 0 ¯ for each b ounded contin uous real function f on the metric space. W e shall sa y then that p U n q and p V n q are “asymptotically equiv alen t”. Note that p µ n q and p ν n q do not necessarily conv erge narrowly to some probabilit y distribution. W e denote by ν n ñ n ν the w eak conv ergence of probabilit y measures. Let f : A Ă X Ñ R . W e deﬁne the function 1 A f by 1 A p x q f p x q “ $ & % f p x q if x P A , 0 else. Denote by D p a, ρ q the op en disk of C with cen ter a P C and radius ρ ą 0, by H the space of holomorphic functions on D p 0 , 1 q , equipp ed with the topology of uniform con vergence on compact subsets of D p 0 , 1 q . It is w ell-known that H is a p olish space. The following conv entions will be used throughout the article: ř H “ 0, ś H “ 1, det p A q “ 1 if A is a matrix of null dimension. F or complex sequences p w n q , p ˜ w n q , the notation u n “ O p v n q implies the existence of EXTREME EIGENV ALUES AND EIGENVECTORS 3 a positive constant κ such that | u n | ď κ | v n | for all n ě 1 suﬃciently large. If we wan t to emphasize the fact that the constan t κ dep ends on some extra parameters z , η , w e may write u n “ O z ,η p v n q . 1.2. Main results. 1.2.1. The mo del. W e b egin b y introducing our random matrix mo del. Let χ b e a complex-v alued random v ariable such that E p χ q “ 0 and E p| χ | 2 q “ 1. F or eac h in teger n ě 1, let A n “ p A n ij q n i,j “ 1 P C n ˆ n b e a random matrix with indep enden t and identically distributed (i.i.d.) elemen ts equal in distribution to χ . Let p K n q b e a sequence of p ositive integers such that K n ď n . Let p B n q b e a sequence of n ˆ n matrices with i.i.d. Bernoulli en tries suc h that, writing B n “ p B n ij q n i,j “ 1 , we hav e P t B n 11 “ 1 u “ K n { n . W e also assume that B n and A n are indep endent. W e consider the sequence of n ˆ n random matrices p X n q n ě 1 giv en as follo ws. W riting X n “ p X n ij q n i,j “ 1 , we set X n ij “ 1 ? K n B n ij A n ij . (1.1) Notice that E X n 11 “ 0 and E | X n 11 | 2 “ 1 { n . Let r ą 0 b e a ﬁxed in teger, and consider 2 r sequences of deterministic v ectors p u 1 ,n q , p u 2 ,n q , ..., p u r,n q , p v 1 ,n q , p v 2 ,n q , ..., p v r,n q such that u t,n , v t,n P C n for each t P r r s and each n ą 0. Consider the sequence p E n q of n ˆ n deterministic matrices deﬁned by E n “ r ÿ t “ 1 u t,n p v t,n q ‹ . W e mak e the follo wing assumptions: Assumption 1. The inte ger se quenc e p K n q satisﬁes K n Ý Ý Ý Ñ n Ñ8 8 . Assumption 2. Ther e exists an absolute c onstant C ą 0 such that r ÿ t “ 1 } u t,n } ` } v t,n } ď C . In man y applicativ e con texts, p K n q con v erges to inﬁnit y at a muc h slo w er pace than n . F or this reason, the parameter K n is referred to as the sparsity parameter of the mo del of X n . Deﬁne the sequence of random matrices p Y n q as Y n “ X n ` E n . It is well-kno wn, see [ SSS25 , Theorem 1.4] whic h generalizes [ R T19 , Theorem 1.2], that the empirical sp ectral distribution of X n con verges to the so-called circular la w. W e shall furthermore sho w in Theorem 1.4 below that the sp ectral radius of X n con verges to 1. In this article, w e study the asymptotic b eha vior of the eigenv alues of Y n whic h Euclidean norm is greater than 1. W e refer to these eigenv alues as outliers , which presence is due to E n . Their b ehavior will b e describ ed in Theorem 1.2 below. In the case of a single outlier, we describ e the b eha vior of the asso ciated eigenv ector. This will b e the conten t of Theorem 1.6 . 1.2.2. Eigenvalues and char acteristic p olynomial of Y n . Our approac h is inspired by the technique developed in [ BCGZ22 ] to capture the asymptotic b ehavior of the sp ectral radius of random matrices with i.i.d. elements, and later extended in [ Cos23 ], [ CLZ23 ], [ FGZ23 ], and [ HL25 ] to v arious other mo dels. One k ey feature of this approac h is that it requires minimal assumptions on the momen ts of the random matrices’ en tries, and it is based on analyzing the asymptotic b ehavior of the rev erse characteristic p olynomial via con vergence to a random analytic function in the unit disk. This latter idea can b e found in [ Shi12 ]. Consider the rev erse characteristic p olynomial of matrix Y n , deﬁned b y (1.2) q n p z q “ det p I n ´ z Y n q . 4 EXTREME EIGENV ALUES AND EIGENVECTORS Clearly , q n is a H -v alued random v ariable. In this pap er, our ﬁrst goal is to study the asymptotic distribution of q n on H . More precisely , we seek an appropriate sequence of random analytic functions φ n P H suc h that q n „ φ n , p n Ñ 8q , where φ n is simpler to analyze that q n . Studying the large- n behavior of q n in the light of the notion of asymptotic equiv alence is well-suited to our purp ose, since without additional assumptions on the matrices E n , there is no reason for p q n q to con verge in law in H . In what follo ws, we deﬁne the sequence of p olynomials p b n q as b n p z q “ det p I ´ z E n q . This sequence is pre-compact in H as a sequence of p olynomials with degrees b ounded b y r and with bounded co eﬃcien ts by Assumption 2 . Theorem 1.1. L et Assumptions 1 and 2 hold true. Consider a se quenc e p Z k q k ě 1 of indep endent Gaussian r andom variables with E p Z k q “ 0 , E p| Z k | 2 q “ 1 , and E p Z 2 k q “ p E A 2 11 q k . Deﬁne κ p z q “ b 1 ´ z 2 E A 2 11 with ? 1 “ 1 , and F p z q “ 8 ÿ k “ 1 z k Z k ? k for z P D p 0 , 1 q . A lso let G n p z q “ b n p z q det p I ´ z X n q . Then q n „ G n , p n Ñ 8q (1.3) as H –value d r andom variables. Also, q n „ b n κ exp p´ F q , p n Ñ 8q , (1.4) as H –value d r andom variables. Pro of of Theorem 1.1 is given in Section 2 . This theorem captures the behavior of the eigen v alues of Y n whic h are a wa y from the unit-disk. In a word, since det p I ´ z Y n q „ det p I ´ z E n q κ p z q exp p´ F p z qq and since the function z ÞÑ κ p z q exp p´ F p z qq has no zero in D p 0 , 1 q , these eigenv alues are close for large n to their counterparts for E n . This is formalized in the next theorem which generalizes Theorem 1.7 of [ T ao13 ] to sparser regimes. W e need the following assumption. Assumption 3. Ther e exists ε ą 0 such that σ p E n q X t z P C : 1 ă | z | ă 1 ` ε u “ H for al l lar ge n . Theorem 1.2. L et Assumptions 1 and 2 hold. Assume that Assumption 3 holds for some ε ą 0 . Deﬁne the set σ ` p E n q “ σ p E n q X t z P C : | z | ą 1 u and σ ` ε p Y n q “ σ p Y n q X t z P C : | z | ě 1 ` ε u and let m n “ | σ ` p E n q| . Then, P ␣ | σ ` ε p Y n q| ‰ m n ( Ý Ý Ý Ñ n Ñ8 0 . F or e ach se quenc e p n 1 q c onver ging to inﬁnity such that m n 1 ą 0 for e ach n 1 , the Hausdorﬀ distanc e b etwe en the sets σ ` ε p Y n 1 q and σ ` p E n 1 q satisﬁes: d H p σ ` ε p Y n 1 q , σ ` p E n 1 qq P Ý Ý Ý Ñ n Ñ8 0 (her e, we set d H pH , σ ` p E n 1 qq “ 8 ). Pr o of of The or em 1.2 given The or em 1.1 . T o prov e the ﬁrst assertion, assume tow ards a contradiction that there exists a sequence p ˜ n q con v erging to inﬁnity such that lim inf n P ␣ | σ ` ε p Y ˜ n q| ‰ m ˜ n ( ą 0. F rom this sequence, extract a subsequence also denoted as p ˜ n q such that b ˜ n con verges to some b 8 in H . Notice that b 8 is a polynomial with a degree b ounded b y r . By Assumption 3 , b 8 has no zero in the ring p 1 ` ε q ´ 1 ă | z | ă 1. Let m 8 ď r be the num b er of zeros of b 8 in D p 0 , 1 q . When m 8 ą 0, let t ζ 1 , ¨ ¨ ¨ , ζ s 8 u b e the set of these zeros not counting m ultiplicities, where s 8 ď m 8 is the num b er of these zeros. In this case, denote as k i the multiplictit y of the EXTREME EIGENV ALUES AND EIGENVECTORS 5 zero ζ i for i P r s 8 s , and deﬁne the set Λ 8 “ t 1 { ζ 1 , ¨ ¨ ¨ , 1 { ζ s 8 u . Then, it holds by , e.g. , Rouch ´ e’s theorem that m ˜ n “ m 8 for all large ˜ n , and furthermore, if m 8 ą 0, that the Hausdorﬀ distance d H p σ ` p E ˜ n q , Λ 8 q conv erges to zero. Indeed, by this theorem, there are k 1 eigen v alues of E ˜ n that conv erge to 1 { ζ 1 , ..., k s 8 eigen v alues of E ˜ n that conv erge to 1 { ζ s 8 , and these eigenv alues exhaust σ ` p E ˜ n q for all large ˜ n . W e shall show that (1.5) | σ ` ε p Y ˜ n q| P Ý Ý Ý Ñ n Ñ8 m 8 , obtaining our con tradiction. By T heorem 1.1 , q ˜ n con verges in la w to w ards the H –v alued random function q 8 p z q “ b 8 p z q κ p z q exp p´ F p z qq . By relying on the explicit expressions of κ and F , notice that function κ exp p´ F q does not v anish on D p 0 , 1 q . If m 8 “ 0, then q 8 do es not v anish on D p 0 , 1 q either. Otherwise, the set of zeros of q 8 coincides with t ζ 1 , ¨ ¨ ¨ , ζ s 8 u with the same multiplicities. By Sk orokho d’s represen tation theorem, there exists a sequence of H –v alued random v ariables p ˇ q ˜ n q and a H –v alued random v ariable ˇ q 8 deﬁned on some common probabilit y space q Ω, such that ˇ q ˜ n law “ q ˜ n , ˇ q 8 law “ q 8 , and ˇ q ˜ n con verges to ˇ q 8 for all ˇ ω P q Ω. W e now ﬁx ˇ ω and apply Rouch ´ e’s theorem. If m 8 “ 0, then ˇ q ˜ n has ev en tually no zero in the compact set t z : | z | ď 1 {p 1 ` ε qu . Otherwise, ˇ q ˜ n has k 1 zeros con v erging to ζ 1 , ¨ ¨ ¨ , k s 8 zeros con v erging to ζ s 8 , and these zeros exhaust the zeros of ˇ q ˜ n in t z : | z | ď 1 {p 1 ` ε qu for all large ˜ n . Getting bac k to q ˜ n , it remains to notice that the zeros of q ˜ n in t z : | z | ď 1 {p 1 ` ε qu , when they exist, are the in verses of the eigenv alues of Y ˜ n in the set t z : | z | ě 1 ` ε u . This establishes the conv ergence ( 1.5 ). The pro of of the second assertion of Theorem 1.2 follows the same canv as. W e just exclude the case where m 8 “ 0. □ T aking E n “ 0, w e obtain the following result. Corollary 1.3. L et Assumption 1 hold and let ρ p X n q b e the sp e ctr al r adius of X n , then for every ε ą 0 , we have P p ρ p X n q ą 1 ` ε q Ý Ý Ý Ñ n Ñ8 0 . Com bining this corollary with the circular law for sparse matrices [ SSS25 , Theorem 1.4], we can generalize [ BCGZ22 , Theorem 1.1] to the sparse case and get: Theorem 1.4. L et Assumption 1 hold and let ρ p X n q b e the sp e ctr al r adius of X n , then ρ p X n q P Ý Ý Ý Ñ n Ñ8 1 . 1.2.3. Eigenve ctors of r ank-one deformation. W e now restrict our atten tion to rank-one deformations. Assuming that r “ 1, write u n “ u 1 ,n and v n “ v 1 ,n for simplicity . The deformation matrix b ecomes then E n “ u n p v n q ‹ . W e need the follo wing assumption: Assumption 4. The deterministic se quenc es p u n q and p v n q satisfy: lim inf n Ñ8 |x v n , u n y| ą 1 . Ob viously , E n is a square n ˆ n matrix whic h only non-zero eigen v alue is x v n , u n y . By the previous assumption, p E n q satisﬁes Assumption 3 , and we imediately hav e the follo wing result: Corollary 1.5 (corollary to Theorem 1.2 ) . L et Assumptions 1 , 2 , and 4 hold true. F or any ﬁxe d ε P p 0 , p lim inf |x v n , u n y| ´ 1 q{ 2 q , c onsider the set σ ` ε p Y n q deﬁne d in the statement of The or em 1.2 . Then, P ␣ | σ ` ε p Y n q| “ 1 ( Ý Ý Ý Ñ n Ñ8 1 . When the event r| σ ` ε p Y n q| “ 1 s is r e alize d, let λ max p Y n q b e the unique eigenvalue of Y n with the lar gest mo dulus, otherwise set λ max p Y n q “ 0 . Then, λ max p Y n q ´ x v n , u n y P Ý Ý Ý Ñ n Ñ8 0 . 6 EXTREME EIGENV ALUES AND EIGENVECTORS In the remainder, when w e men tion the ev ent r| σ ` ε p Y n q| “ 1 s , we assume that ε ą 0 is small enough according to the statement of the previous corollary . Our ob jective is to analyze the pro jection on u n of the right eigen vector of Y n corresp onding to λ max p Y n q (assuming r| σ ` ε p Y n q| “ 1 s is realized). Our main tec hnical to ols are based on the results from [ BvH24 ], which allo w us to compare the sp ectral prop erties of X n with a Gaussian analogue to this matrix. W e will need the following extra sub-Gaussian assumption concerning A n ’s entries. Assumption 5 (sub-Gaussiannit y) . The r andom variables A ij fol low a sub-Gaussian distribution, i.e. , ther e exists an absolute c onstant C ą 0 such that P p| A n 11 | ě t q ď 2 exp p´ C t 2 q . W e are now in position to describe the eigenv ectors of Y n “ X n ` u n p v n q ‹ corresp onding to the outlier λ max p Y n q . Theorem 1.6. L et Assumptions 1 , 2 , 4 and 5 hold true. Assume furthermor e that lim n Ñ8 log 9 n K n “ 0 . When the event t| σ ` ε p Y n q| “ 1 u is r e alize d, let ˜ u n b e an unit-norm right eigenve ctor of Y n c orr esp onding to λ max p Y n q . Otherwise, put ˜ u n “ 0 n . Then, it holds that ˇ ˇ ˇ ˇ B ˜ u n , u n } u n } F ˇ ˇ ˇ ˇ 2 ´ ˆ 1 ´ 1 |x u n , v n y| 2 ˙ P Ý Ý Ý Ñ n Ñ8 0 . Pro of of Theorem 1.6 is p ostp oned to Section 4 . R emark 1.7 . In the case where u n is a unit-norm v ector and where one considers the model Y n “ X n ` αu n p u n q ‹ for some ﬁxed α ą 1, then the result ab ov e b oils down to |x ˜ u n , u n y| 2 P Ý Ý Ý Ñ n Ñ8 1 ´ 1 α 2 . In terestingly , this corresp onds to the same quantit y as in the Hermitian case, see [ BGN11 , Section 3.1]. 2. Proof of Theorem 1.1 This section is devoted to the pro of of Theorem 1.1 . W e follow the strategy developed in [ BCGZ22 ]. 2.1. Tigh tness and truncation. W e ﬁrst state useful prop erties for H -v alued random v ariables. Prop osition 2.1 (Tigh tness criterion [ HL25 , Prop osition 3.1]) . L et p f n q b e a se quenc e of H -value d r andom variables. If for every c omp act set K Ă D p 0 , 1 q , sup n sup z P K E | f n p z q| 2 ď C K ă 8 , for some K -dep endent c onstant C K , then p f n q is tight. Prop osition 2.2 (Asymptotic equiv alence criteria in H ) . L et p f n q and p g n q b e two tight se quenc es of H - value d r andom variables. Consider their p ower series r epr esentations in D p 0 , 1 q : f n p z q “ ř 8 k “ 0 a p n q k z k and g n p z q “ ř k ě 0 b p n q k z k . If one of the fol lowing c onditions holds: (1) F or every ﬁxe d inte ger m ě 1 , p a p n q 0 , ¨ ¨ ¨ , a p n q m q „ n p b p n q 0 , ¨ ¨ ¨ , b p n q m q , (2) F or every ﬁxe d inte ger m ě 1 and m -uple p z 1 , ¨ ¨ ¨ z m q P D m p 0 , 1 q , p f n p z 1 q , ¨ ¨ ¨ , f n p z m qq „ n p g n p z 1 q , ¨ ¨ ¨ , g n p z m qq , then f n „ g n as n Ñ 8 . Most of the time, w e shall drop the dep endence in n for notational conv enience. Prop osition 2.3 (Tightness) . L et Assumptions 1 and 2 hold. L et q n b e given by ( 1.2 ) , then the se quenc e p q n q n ě 1 is tight in H . EXTREME EIGENV ALUES AND EIGENVECTORS 7 Pr o of. W e ﬁrst recall a w ell-kno wn general result. Let A and B b e t w o n ˆ n matrices with columns A i and B i resp ectiv ely for i P r n s . Then, using the multilinearit y of the determinant, w e can write det p A ` B q “ det ” A 1 ` B 1 A 2 ` B 2 ¨ ¨ ¨ A n ` B n ı “ det ” A 1 A 2 ` B 2 ¨ ¨ ¨ A n ` B n ı ` det ” B 1 A 2 ` B 2 ¨ ¨ ¨ A n ` B n ı “ det ” A 1 A 2 ¨ ¨ ¨ A n ` B n ı ` det ” A 1 B 2 ¨ ¨ ¨ A n ` B n ı ` ¨ ¨ ¨ whic h will ultimately provide a “binomial-like” expression of det p A ` B q that will hav e the following form. Giv en k P t 0 , . . . , n u , let I P r n s with | I | “ k and all the elemen ts of I are diﬀeren t, and denote as p A, B q I the n ˆ n matrix which i th column is A i if i P I and B i if i P r n sz I . Then, (2.1) det p A ` B q “ n ÿ k “ 0 ÿ I Pr n s : | I |“ k det p A, B q I . Let us write M “ I ´ z E “ r M ij s n i,j “ 1 , so that q p z q “ det p´ z X ` M q . W riting E | q p z q| 2 “ ÿ σ, ˜ σ P S n E p´ z X 1 ,σ p 1 q ` M 1 ,σ p 1 q q . . . p´ z X n,σ p n q ` M n,σ p n q qp´ ¯ z ¯ X 1 , ˜ σ p 1 q ` ¯ M 1 , ˜ σ p 1 q q . . . p´ ¯ z ¯ X n, ˜ σ p n q ` ¯ M n, ˜ σ p n q q , w e see that the element X ij acts on E | q p z q| 2 through E X ij and E | X ij | 2 only . Therefore, E | q p z q| 2 is inv arian t if w e assume that these elements are i.i.d. with X 11 „ N C p 0 , 1 { n q , which w e do from now on in this pro of. Denoting as M “ U Σ V ‹ a singular v alue decomposition of M , we ha v e | q p z q| 2 “ det p´ z X ` M qp´ ¯ z X ‹ ` M ‹ q “ det p´ z U ‹ X V ` Σ qp´ ¯ z V ‹ X ‹ U ` Σ q L “ | det p z X ` Σ q| 2 . W e also ha v e that the matrix M M ‹ “ I ´ z E ´ ¯ z E ‹ ` | z | 2 E E ‹ is equal to the iden tit y plus a deformation of rank 2 r at most. Therefore, the diagonal n ˆ n matrix Σ of the singular v alues of M contains ones on its diagonal except for 2 r singular v alues at most. Moreo v er, using Assumption 2 and recalling that z P D p 0 , 1 q , w e obtain that there exists C Σ ě 1 indep endent of n and z such that } Σ } ď C Σ . W e now compute E | q p z q| 2 “ E | det p z X ` Σ q| 2 where we develop det p z X ` Σ q using the formula ( 2.1 ). Here, w e can notice that E det p z X , Σ q I det p z X , Σ q ˜ I “ 0 if I ‰ ˜ I . Indeed, the case b eing, one of the matrices p z X , Σ q I or p z X , Σ q ˜ I con tains a column of z X that is not present in the other. Making a Laplace expansion of the corresponding determinant along this column, we obtain that the cross exp ectation is zero. W e therefore get that E | q p z q| 2 “ n ÿ k “ 0 ÿ I Pr n s : | I |“ k E | det p z X , Σ q I | 2 . Let us work on one of these determinants. F or a given k , let us assume for simplicity that I “ r k s . Otherwise, w e can p ermute the ro ws and columns of p z X , Σ q I prop erly; this do es not aﬀect | det p z X , Σ q I | 2 . W riting r k s c “ r n szr k s , we ha v e det p z X , Σ q I “ det p z X , Σ q r k s “ det « z X r k s , r k s 0 z X r k s c , r k s Σ r k s c , r k s c ﬀ “ z k det X r k s , r k s det Σ r k s c , r k s c , and E | det p z X , Σ q r k s | 2 “ | z | 2 k ˇ ˇ det Σ r k s c , r k s c ˇ ˇ 2 E | det X r k s , r k s | 2 . By the prop erties of Σ stated ab ov e, we ha v e ˇ ˇ det Σ r k s c , r k s c ˇ ˇ 2 ď C 4 r Σ . Moreov er, if k “ 0, then E | det X r k s , r k s | 2 “ 1, otherwise, E | det X r k s , r k s | 2 “ ÿ σ P S k E ˇ ˇ X 1 ,σ p 1 q . . . X k,σ p k q ˇ ˇ 2 “ k ! n k . Therefore, E | det p z X , Σ q r k s | 2 ď | z | 2 k C 4 r Σ k ! { n k , and w e end up with E | q p z q| 2 “ n ÿ k “ 0 ÿ I Pr n s : | I |“ k E | det p z X , Σ q I | 2 ď C 4 r Σ n ÿ k “ 0 ˆ n k ˙ k ! n k | z | 2 k ď C 4 r Σ 8 ÿ k “ 0 | z | 2 k “ C 4 r Σ 1 ´ | z | 2 . The tightness of p q n q follows by applying Prop osition 2.1 . □ Moreo ver it is suﬃcient to examine the characteristic p olynomial of Y n , when the entries of A n are b ounded almost surely . Sp eciﬁcally 8 EXTREME EIGENV ALUES AND EIGENVECTORS Prop osition 2.4 (T runcation) . L et Assumptions 1 and 2 hold. L et D ą 0 and deﬁne A D i,j “ A i,j 1 | A i,j |ď D ´ E A i,j 1 | A i,j |ď D , X n,D i,j “ 1 ? K n B n i,j A D i,j and Y n,D ij “ X n,D ij ` E n ij , p i, j P r n sq . L et X n,D “ ” X n,D i,j ı i,j Pr n s , Y n,D “ X n,D ` E n and q D n p z q “ det ` I n ´ z Y n,D ˘ . Then, @ z P D p 0 , 1 q , sup n E ˇ ˇ q n p z q ´ q D n p z q ˇ ˇ 2 ď ε p D q wher e ε p D q Ý Ý Ý Ý Ñ D Ñ8 0 . Pr o of. W e omit the sup ersript n in the sequel to lighten the notations. Without a risk of confusion, we replace, e.g. , Y n,D with Y D . W e closely follow the principles and notations introduced in the previous pro of. Let M “ I ´ z E as ab ov e. W riting E ˇ ˇ q n p z q ´ q D n p z q ˇ ˇ 2 “ E ˇ ˇ ˇ ˇ ˇ ÿ σ P S n ˜ n ź i “ 1 p´ z X i,σ p i q ` M i,σ p i q q ´ n ź i “ 1 p´ z X D i,σ p i q ` M i,σ p i q q ¸ ˇ ˇ ˇ ˇ ˇ 2 and developing, w e notice that E ˇ ˇ q n p z q ´ q D n p z q ˇ ˇ 2 dep ends on each element X ij via the vector E « X ij X D ij ﬀ p“ 0 q and the 2 ˆ 2 matrix R D “ n E « X ij X D ij ﬀ ” X ij X D ij ı whic h do es not dep end on n . Therefore, we can assume without loss of generality that the vector ? n « X ij X D ij ﬀ is a cicularly symmetric Gaussian vector (see the deﬁnition in [ T el99 ] for instance) with cov ariance matrix R D , and in particular: E « X ij X D ij ﬀ “ 0 , n E « X ij X D ij ﬀ ” X ij X D ij ı “ R D and n E « X ij X D ij ﬀ ” X ij X D ij ı “ 0 . Assuming this, we ﬁrst observ e that vec r X X D s is a C 2 n 2 –v alued circularly symmetric Gaussian vector, and so is vector A vec r X X D s for an y deterministic p ˆ 2 n 2 matrix A . Consider now n ˆ n deterministic matrices U, V . Applying [ HJ94 , Lemma 4.3.1] w e hav e v ec r U X V U X D V s “ p V T b U q vec r X X D s , hence vec r U X V U X D V s is circularly symmetric Gaussian, in particular E r U X V s ij r U Y V s st “ 0 for any i, j, s, t P r n s and Y P t X , X D u . W e now wish to understand the cov ariance structure of the comp onents of vec r U X V U X D V s in the case where U and V are unitary . E r U X V s ij r U X V s st “ ÿ k,ℓ ÿ p,q U ik U sp V ℓj V q t E X kℓ X pq , “ ÿ k U ik U sk ÿ ℓ V ℓj V ℓt r R D s 11 , “ r U U ‹ s is r V ‹ V s tj r R D s 11 “ δ is δ j t r R D s 11 where δ ab “ $ & % 1 if a “ b 0 else . Similarly we can prov e that E r U X V s ij r U X D V s st “ δ is δ j t r R D s 12 and E r U X D V s ij r U X D V s st “ δ is δ j t r R D s 22 . Collecting all these prop erties, w e hav e prov ed that for any n ˆ n deterministic, unitary matrices U, V , r X X D s L “ r U X V U X D V s . EXTREME EIGENV ALUES AND EIGENVECTORS 9 Therefore, using the singular v alue decomposition M “ U Σ V ‹ and Equation ( 2.1 ), we hav e: E ˇ ˇ q n p z q ´ q D n p z q ˇ ˇ 2 “ E ˇ ˇ det p´ z U ‹ X V ` Σ q ´ det p´ z U ‹ X D V ` Σ q ˇ ˇ 2 “ E ˇ ˇ det p z X ` Σ q ´ det p z X D ` Σ q ˇ ˇ 2 “ E ˇ ˇ ˇ ˇ ˇ ˇ n ÿ k “ 0 ÿ I Ăr n s : | I |“ k det p z X , Σ q I ´ det p z X D , Σ q I ˇ ˇ ˇ ˇ ˇ ˇ 2 “ n ÿ k “ 0 ÿ I Ăr n s : | I |“ k E ˇ ˇ det p z X , Σ q I ´ det p z X D , Σ q I ˇ ˇ 2 , b y relying on the fact (established in the previous pro of ) that E det p z X , Σ q I det p z X D , Σ q ˜ I “ 0 if I ‰ ˜ I . Let I “ r k s as ab ov e for some k ě 1, then E ˇ ˇ det p z X , Σ q I ´ det p z X D , Σ q I ˇ ˇ 2 “ | z | 2 k E ˇ ˇ ˇ det X r k s , r k s ´ det X D r k s , r k s ˇ ˇ ˇ 2 ˇ ˇ det Σ r k s c , r k s c ˇ ˇ 2 , ď | z | 2 k C p Σ , r q E ˇ ˇ ˇ det X r k s , r k s ´ det X D r k s , r k s ˇ ˇ ˇ 2 , where C p Σ , r q is a constant indep endent of n b y Assumption 2 . Recall that E | A 11 | 2 “ 1, notice that E | A D 11 | 2 ď 1 and ε p D q : “ E | A 11 ´ A D 11 | 2 Ý Ý Ý Ý Ñ D Ñ8 0 . W e ha v e: E ˇ ˇ ˇ ˇ ˇ ˇ ź i Pr k s X 1 i ´ ź i Pr k s X D 1 i ˇ ˇ ˇ ˇ ˇ ˇ 2 “ E ˇ ˇ p X 11 ´ X D 11 q X 12 ¨ ¨ ¨ X 1 k ` ¨ ¨ ¨ ` X D 11 ¨ ¨ ¨ X D 1 ,k ´ 1 p X 1 k ´ X D 1 k q ˇ ˇ 2 ď k n k ε p D q b y Minko wski’s inequality . W e therefore obtain that E ˇ ˇ ˇ det X r k s , r k s ´ det X D r k s , r k s ˇ ˇ ˇ 2 “ E ˇ ˇ ˇ ˇ ˇ ÿ σ S k X 1 σ p 1 q ¨ ¨ ¨ X kσ p k q ´ X D 1 σ p 1 q ¨ ¨ ¨ X D kσ p k q ˇ ˇ ˇ ˇ ˇ 2 ď k ! k n k ε p D q . No w, ÿ I Ăr n s : | I |“ k E ˇ ˇ det p z X , Σ q I ´ det p z X D , Σ q I ˇ ˇ 2 ď ˆ n k ˙ k ! k n k ε p D q , and ﬁnally E ˇ ˇ q n p z q ´ q D n p z q ˇ ˇ 2 ď C p Σ , r q ε p D q 8 ÿ k “ 0 k | z | 2 k ď C ε D . The prop osition is prov en. □ 2.2. Momen ts of Y n and X n . W e study the asymptotic b ehavior of the vector ` 1 , tr p Y n q , . . . , tr pp Y n q k q ˘ , k P N . Throughout, we write tr pp Y n q k q (and similarly for X n , E n ); this is the quan tity expanded b elow. Circles : W e consider directed circles (called cir cles ) consisting of exactly k edges. In our setting, a cir cle is an Eulerian cycle of a strongly connected directed multigraph; v ertices ma y rep eat and multiple edges (including lo ops and parallel edges) are allo w ed. W e identify underlying strongly connected directed m ultigraphs up to graph isomorphism, and denote by C k the collection of all Eulerian cycles of length k arising from all suc h isomorphism classes. F ormally , an elemen t C P C k can b e represen ted by a cyclic sequence C “ t u 1 , u 2 , . . . , u k , u 1 u , 10 EXTREME EIGENV ALUES AND EIGENVECTORS where the vertices u i are not necessarily distinct, and eac h consecutiv e pair p u i , u i ` 1 q forms a directed edge (with the con ven tion u k ` 1 “ u 1 ). W e denote by V p C q the set of vertices app earing in C , and by E p C q “ ␣ p u i , u i ` 1 q : i “ 1 , . . . , k ( the multiset of edges. F or an edge e “ p u, v q P E p C q , its multiplicity is ˇ ˇ t ˜ e P E p C q : ˜ e “ e u ˇ ˇ . W e call u the sour c e of e and v its tar get . F or any C P C k and B Ă E p C q , we denote by C z B the directed m ultigraph with edge m ultiset E p C qz B and v ertex set induced by these edges. Lab elings. Giv en B Ă E p C q and a labeling i P r n s | V p C q| , we write i „ C if i assigns distinct indices from r n s to the v ertices of C according to their ﬁrst order of appearance along the circle. W e denote b y i p B q the (m ulti)set of lab eled edges corresp onding to B . With this notation, tr ` p Y n q k ˘ “ ÿ p i 1 ,...,i k qPr n s k k ź ℓ “ 1 p X n ` E n q i ℓ ,i ℓ ` 1 “ ÿ C P C k ÿ B Ă E p C q ÿ i „ C i Pr n s | V p C q| ź p i,j qP i p B q E n i,j ź p i,j qR i p B q X n i,j , under the con ven tion i k ` 1 “ i 1 . Therefore, tr ` p Y n q k ˘ ´ tr ` p X n q k ˘ ´ tr ` p E n q k ˘ “ ÿ C P C k ÿ B Ă E p C q B ‰H , E p C q ÿ i „ C i Pr n s | V p C q| ź p i,j qP i p B q E n i,j ź p i,j qR i p B q X n i,j . (2.2) A uxiliary notation. Notation 1. ‚ F or any ﬁnite multiset A , we denote by | A | no the c ar dinality of its underlying set (i.e., ignoring multiplicities). ‚ F or C P C k and B Ă E p C q , deﬁne E ` b d p C z B q ˘ “ ! e P B : D v P V p C z B q such that v is incident to e ) . Then b d p C z B q denotes the dir e cte d multigr aph induc e d by the e dge multiset E p b d p C z B qq . ‚ F or C P C k and B Ă E p C q , let C B b e the dir e cte d multigr aph induc e d by the e dge multiset B . We write i „ C B to indic ate a lab eling i P r n s | V p C B q| assigning distinct values to the vertic es of C B . ‚ F or any dir e cte d multigr aph G and v P V p G q , we denote by deg ` G p v q (r esp. deg ´ G p v q ) the out-de gr e e (r esp. in-de gr e e) of v , c ounte d with multiplicity. A c ombinatorial ine quality. Lemma 2.5. Fix C P C k and B Ă E p C q such that B ‰ H and B ‰ E p C q . Assume: (1) C z B is we akly c onne cte d; (2) for every e P E p C z B q , the multiplicity |t ˜ e P E p C q : ˜ e “ e u| ě 2 . Then | V p C z B q| ´ ˇ ˇ ␣ v P V p C z B q X V p b d p C z B qq : deg ` C B p v q ` deg ´ C B p v q ě 2 ( ˇ ˇ ´ 1 2 ˇ ˇ ␣ v P V p C B q : deg ` C B p v q ` deg ´ C B p v q “ 1 ( ˇ ˇ ´ | E p C z B q| no ď ´ 1 . Pr o of. Since C z B is w eakly connected, one has the standard b ound | V p C z B q| ď | E p C z B q| no ` 1 . W e distinguish the follo wing cases: (1) | V p C z B q| “ | E p C z B q| no ` 1; EXTREME EIGENV ALUES AND EIGENVECTORS 11 (2) | V p C z B q| “ | E p C z B q| no and ˇ ˇ ␣ v P V p C z B q X V p b d p C z B qq : deg ` C B p v q ` deg ´ C B p v q ě 2 ( ˇ ˇ “ 0; (3) | V p C z B q| “ | E p C z B q| no and ˇ ˇ ␣ v P V p C z B q X V p b d p C z B qq : deg ` C B p v q ` deg ´ C B p v q ě 2 ( ˇ ˇ ě 1; (4) | V p C z B q| ă | E p C z B q| no . Cases 3 and 4 are immediate from the deﬁnition of the left-hand side, so w e treat 1 and 2 . Case 1 . Here the underlying simple undirected graph of C z B is a tree. Deﬁne V ` p C z B q “ t v P V p C z B q : D p v , a q P E p C z B qu , V ´ p C z B q “ t v P V p C z B q : D p a, v q P E p C z B qu . Since C z B is a ﬁnite tree, there exist v ertices w R V ` p C z B q and u R V ´ p C z B q ; otherwise, starting from any v ertex one could construct an inﬁnite directed path, contradicting ﬁniteness. Each of u, w is inciden t to at least one edge of C z B , and b y assumption those edges ha v e multiplicit y at least 2 in C . Because C is a circle, there are at least t wo directed edges in C leaving w and at least t wo entering u . By the choice of u, w , these additional edges must b elong to B , yielding the required inequality . Case 2 . Since C is a circle and no b oundary v ertex has total degree at least 2 in C B , we hav e | V p C z B q X V p b d p C z B qq| “ ˇ ˇ ␣ v P V p C B q : deg ` C B p v q ` deg ´ C B p v q “ 1 ( ˇ ˇ (2.3) “ | E p bd p C z B qq| ě 2 . Indeed, if ( 2.3 ) failed, then C either could not en ter or could not exit C z B , con tradicting that C is a circle. The claim follo ws. □ R emark 2.6 . If C z B has several weakly connected comp onents, Lemma 2.5 applies to each comp onent separately . Main c ombinatorial c onse quenc e. Prop osition 2.7. Assume that | A n 1 , 1 | ď D for some (ﬁxe d) c onstant D ą 0 . Then for every k P N , tr ` p Y n q k ˘ ´ tr ` p X n q k ˘ ´ tr ` p E n q k ˘ P Ý Ý Ý Ñ n Ñ8 0 . Pr o of. W e pro v e the claim by controlling the mean and the v ariance. Step 1: b ound on the mean. W e sho w that E ” tr ` p Y n q k ˘ ´ tr ` p X n q k ˘ ´ tr ` p E n q k ˘ ı “ O ˆ 1 n ˙ . (2.4) By ( 2.2 ), it suﬃces to bound ÿ C P C k ÿ B Ă E p C q B ‰ E p C q , H ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ E ź p i,j qR i p B q X n i,j ˇ ˇ ˇ ˇ ˇ ˇ “ O ˆ 1 n ˙ . (2.5) F or the exp ectation in ( 2.5 ) to b e non-zero, every edge of C z B must ha v e multiplicit y at least 2. Since the n um b er of circles in C k and the n um b er of subsets B Ă E p C q dep end only on k , it is enough to ﬁx C P C k and a non-trivial B Ă E p C q and prov e ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ E ź p i,j qR i p B q X n i,j ˇ ˇ ˇ ˇ ˇ ˇ “ O ˆ 1 n ˙ , (2.6) under the assumption that every edge in C z B has multiplicit y at least 2. W e also assume for simplicity that C z B is weakly connected (the case of several w eakly connected comp onen ts is treated comp onent-wise). 12 EXTREME EIGENV ALUES AND EIGENVECTORS Since the en tries of A n are b ounded b y D , we obtain ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ E ź p i,j qR i p B q X n i,j ˇ ˇ ˇ ˇ ˇ ˇ ď D 2 k ˆ 1 ? K n ˙ | E p C z B q| ˆ K n n ˙ | E p C z B q| no ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ . Since each edge of C z B app ears at least t wice, we ha ve | E p C z B q| no ď | E p C z B q| 2 . (2.7) Moreo ver, connectivity of the underlying simple graph yields | V p C z B q| ď | E p C z B q| no ` 1 . (2.8) It remains to b ound ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ . Using the represen tation of E n and the en trywise b ound | E n i,j | ď r max ℓ Pr r s | u ℓ,n i | max ℓ Pr r s | v ℓ,n j | , w e obtain (as in ( 2.9 ) in the original deriv ation) ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ ď r k n | V p C z B q|´| V p C z B qX V p b d p C z B qq| ź v P V p C B q ¨ ˝ ÿ i Pr n s max ℓ Pr r s | v ℓ,n i | deg ´ C B p v q max ℓ Pr r s | u ℓ,n i | deg ` C B p v q ˛ ‚ . (2.9) No w, for each v P V p C B q : ‚ If deg ` C B p v q ` deg ´ C B p v q ě 2, then by Assumption 2 (and the same Cauch y–Sc h warz argument as in the original pro of ), ÿ i Pr n s max ℓ Pr r s | v ℓ,n i | deg ´ C B p v q max ℓ Pr r s | u ℓ,n i | deg ` C B p v q ď C for some constan t C ą 0. ‚ If deg ` C B p v q ` deg ´ C B p v q “ 1, then by Cauch y–Sch w arz, ÿ i Pr n s max ℓ Pr r s | v ℓ,n i | deg ` C B p v q max ℓ Pr r s | u ℓ,n i | deg ´ C B p v q ď r C ? n. Th us, for some C “ C p k , r q ą 0, ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ ď C n | V p C z B q|´| V p C z B qX V p b d p C z B qq|` 1 2 ˇ ˇ ˇ t v P V p C B q :deg ` C B p v q` deg ´ C B p v q“ 1 u ˇ ˇ ˇ . Because C is a circle, any vertex v with deg ` C B p v q “ 0 or deg ´ C B p v q “ 0 m ust lie in V p C z B q X V p b d p C z B qq . Hence, setting a p C z B q : “ | V p C z B q| ´ ˇ ˇ ␣ v P V p C z B q X V p b d p C z B qq : deg ` C B p v q ` deg ´ C B p v q ě 2 ( ˇ ˇ ´ 1 2 ˇ ˇ ␣ v P V p C B q : deg ` C B p v q ` deg ´ C B p v q “ 1 ( ˇ ˇ , EXTREME EIGENV ALUES AND EIGENVECTORS 13 w e conclude that for a constant C k indep enden t of n , ÿ i „ C i Pr n s | V p C q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ E ź p i,j qR i p B q X n i,j ˇ ˇ ˇ ˇ ˇ ˇ ď C k p K n q | E p C z B q| no ´ | E p C z B q| 2 n a p C z B q´| E p C z B q| no . (2.10) The desired O p 1 { n q b ound follows from Lemma 2.5 together with ( 2.7 ). This prov es ( 2.4 ). Step 2: b ound on the v ariance. W e show that V ar ´ tr ` p Y n q k ˘ ´ tr ` p X n q k ˘ ´ tr ` p E n q k ˘ ¯ “ O ˆ 1 nK n ˙ . (2.11) Recall that for complex random v ariables t W i u m i “ 1 , V ar ´ m ÿ i “ 1 W i ¯ “ ÿ i 1 ,i 2 Pr m s E ” ` W i 1 ´ E W i 1 ˘` W i 2 ´ E W i 2 ˘ ı . Applying this to the expansion ( 2.2 ) yields V ar ´ tr ` p Y n q k ˘ ´ tr ` p X n q k ˘ ´ tr ` p E n q k ˘ ¯ “ ÿ C , C 1 P C k ÿ B Ă E p C q B ‰H , E p C q ÿ B 1 Ă E p C 1 q B 1 ‰H , E p C 1 q ÿ i „ C i Pr n s | V p C q| ÿ i 1 „ C 1 i 1 Pr n s | V p C 1 q| ¨ ź p i,j qP i p B q E n i,j ź p i 1 ,j 1 qP i 1 p B 1 q E n i 1 ,j 1 E ´ ź p i,j qR i p B q X n i,j ´ E ź p i,j qR i p B q X n i,j ¯ ˆ ´ ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ´ E ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ¯ . (2.12) By indep endence of the entries of X n , the exp ectation in ( 2.12 ) v anishes unless the lab eled edge sets in C z B and C 1 z B 1 agree on at least one edge, and ev ery edge in the multigraph C z B Y C 1 z B 1 has multiplicit y at least 2. Hence, it suﬃces to show that for any suc h C , C 1 and non-trivial B , B 1 , ÿ i „ C i Pr n s | V p C q| ÿ i 1 „ C 1 i 1 Pr n s | V p C 1 q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ź p i 1 ,j 1 qP i 1 p B 1 q E n i 1 ,j 1 ˇ ˇ ˇ ˇ ˇ ˇ ¨ ˆ ˇ ˇ ˇ ˇ ˇ ˇ E ´ ź p i,j qR i p B q X n i,j ´ E ź p i,j qR i p B q X n i,j ¯´ ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ´ E ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ¯ ˇ ˇ ˇ ˇ ˇ ˇ “ O ˆ 1 nK n ˙ . (2.13) F or simplicity , assume C z B and C 1 z B 1 are weakly connected; the general case follows by decomp osing in to w eakly connected comp onents. Since C z B and C 1 z B 1 share an edge, the union C z B Y C 1 z B 1 is also weakly connected. Set r C : “ C Y C 1 . After an appropriate ordering of v ertices, r C deﬁnes a circle of length 2 k . Let r C B denote the subgraph induced by the edge multiset B Y B 1 . Using the deﬁnition of X n (cf. ( 1.1 )) and the indep endence structure, together with the b ound | A n ij | ď D , we ha ve ˇ ˇ ˇ ˇ ˇ ˇ E ´ ź p i,j qR i p B q X n i,j ´ E ź p i,j qR i p B q X n i,j ¯´ ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ´ E ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ¯ ˇ ˇ ˇ ˇ ˇ ˇ ď 2 D 2 k ˆ K n n ˙ | E p C z B Y C 1 z B 1 q| no . Moreo ver, as in ( 2.9 ), one shows that ÿ i „ C i Pr n s | V p C q| ÿ i 1 „ C 1 i 1 Pr n s | V p C 1 q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP i p B q E n i,j ź p i 1 ,j 1 qP i 1 p B 1 q E n i 1 ,j 1 ˇ ˇ ˇ ˇ ˇ ˇ ď C n a p C z B , C 1 z B 1 q , 14 EXTREME EIGENV ALUES AND EIGENVECTORS where a p C z B , C 1 z B 1 q “ | V p C z B q Y V p C 1 z B 1 q| ´ ˇ ˇ ˇ ! v : deg ` r C B p v q ` deg ´ r C B p v q ě 2 ) ˇ ˇ ˇ ´ 1 2 ˇ ˇ ˇ ! v : deg ` r C B p v q ` deg ´ r C B p v q “ 1 ) ˇ ˇ ˇ . Since r C is a circle and every edge in C z B Y C 1 z B 1 has multiplicit y at least 2, Lemma 2.5 applies and yields (for some C k dep ending only on k , M , r ) ( 2.13 ) ď C k n ¨ 1 b K | E p C z B q|`| E p C 1 z B 1 q| n ¨ K | E p C z B Y C 1 z B 1 q| no n . Finally , since all edges in C z B Y C 1 z B 1 app ear at least twice and the tw o graphs share an edge, w e hav e | E p C z B Y C 1 z B 1 q| no ď 2 ` | E p C z B q| ` | E p C 1 z B 1 q| ´ 1 ˘ , and therefore ( 2.13 ) ď C k 1 nK n . This prov es ( 2.11 ). Com bining ( 2.4 ) and ( 2.11 ) gives tr pp Y n q k q ´ tr pp X n q k q ´ tr pp E n q k q Ñ 0 in probabilit y , completing the pro of. □ W e contin ue with the asymptotic analysis of the joint law of the traces tr pp X n q k q . Recall the notation from Theorem 1.1 and deﬁne, for k P N , the sequence mean k : “ 1 t k even u ` E A 2 1 , 1 ˘ k { 2 . Lemma 2.8. F or any k ě 1 , if | A n 1 , 1 | ď D almost sur ely, then ` tr p X n q , . . . , tr pp X n q k q ˘ law Ý Ý Ý Ñ n Ñ8 ` Z 1 ` mean 1 , ? 2 Z 2 ` mean 2 , . . . , ? k Z k ` mean k ˘ . Pr o of. When K n ě log n , the claim follows directly from Propositions 2.3 and 3.6 of [ HL25 ], applied to our mo del. In general, the pro of is analogous to that of Lemmas 3.4 and 3.5 in [ BCGZ22 ]. F or completeness, we sketc h the main steps. Recall the notation C k for the collection of directed circles of length k . Let k 1 , . . . , k m P N , let C ℓ P C k ℓ for ℓ “ 1 , . . . , m , and let s 1 , . . . , s m P t¨ , ˚u , where for any complex n um b er x we set x ¨ “ x and x ˚ “ x . Deﬁne the m ultigraph r C : “ m ď ℓ “ 1 C ℓ . Then the join t contribution of these circles satisﬁes ÿ i „ r C ˇ ˇ ˇ ˇ ˇ ˇ E m ź ℓ “ 1 ź p v ,u qP E p C ℓ q ` X n i p v q , i p u q ˘ s ℓ ˇ ˇ ˇ ˇ ˇ ˇ ď D ř m ℓ “ 1 k ℓ K | E p r C q| no ´ 1 2 ř m ℓ “ 1 | E p C ℓ q| n n | V p r C q|´| E p r C q| no . (2.14) Since the en tries of X n are centered, the contribution in ( 2.14 ) is negligible unless | V p r C q| “ | E p r C q| no and 2 | E p r C q| no “ m ÿ ℓ “ 1 | E p C ℓ q| . W e pro ceed as in [ BCGZ22 ]. Decomp ose C k as C k “ C 1 k Y C 2 k , where C 1 k consists of circles with exactly k distinct vertices, and C 2 k consists of circles with few er than k vertices. Accordingly , we write tr pp X n q k q “ ÿ C P C 1 k ÿ i „ C i Pr n s k ź p v ,u qP E p C q X n i p v q , i p u q ` ÿ C P C 2 k ÿ i „ C i Pr n s k ź p v ,u qP E p C q X n i p v q , i p u q “ : t k n ` r k n . The pro of is complete once we establish the following t w o facts. EXTREME EIGENV ALUES AND EIGENVECTORS 15 (1) F or an y k 1 , . . . , k m P N and s 1 , . . . , s m P t¨ , ˚u , E m ź ℓ “ 1 p t k ℓ n q s ℓ Ý Ý Ý Ñ n Ñ8 E m ź ℓ “ 1 ` a k ℓ Z k ℓ ˘ s ℓ . (2.15) (2) F or an y k P N , r k n P Ý Ý Ý Ñ n Ñ8 mean k . (2.16) Giv en the b ound ( 2.14 ), the con v ergence ( 2.15 ) and ( 2.16 ) follow exactly as in the proofs of Lemmas 3.4 and 3.5 of [ BCGZ22 ], resp ectiv ely . □ W e conclude with an asymptotic b ound on E | tr pp Y n q k q| 2 , which is needed to establish relative compactness. Lemma 2.9. F or any k P N , if | A n 1 , 1 | ď D , then ther e exists a c onstant C “ C p r, k q ą 0 such that E ˇ ˇ tr pp Y n q k q ˇ ˇ 2 ď C . Pr o of. W e begin with the expansion E ˇ ˇ tr pp Y n q k q ˇ ˇ 2 “ ÿ C , C 1 P C k ÿ B Ă E p C q B ‰H ,E p C q ÿ B 1 Ă E p C 1 q B 1 ‰H ,E p C 1 q ÿ i „ C i Pr n s | V p C q| ÿ i 1 „ C 1 i 1 Pr n s | V p C 1 q| ¨ ¨ ¨ ź p i,j qP B E n i,j ź p i 1 ,j 1 qP B 1 E n i 1 ,j 1 E » – ź p i,j qR i p B q X n i,j ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ﬁ ﬂ . (2.17) W e pro ceed as in the pro of of Prop osition 2.7 . Fix C , C 1 P C k and non-trivial subsets B Ă E p C q , B 1 Ă E p C 1 q . It suﬃces to show that ÿ i „ C i Pr n s | V p C q| ÿ i 1 „ C 1 i 1 Pr n s | V p C 1 q| ˇ ˇ ˇ ˇ ˇ ˇ ź p i,j qP B E n i,j ź p i 1 ,j 1 qP B 1 E n i 1 ,j 1 E » – ź p i,j qR i p B q X n i,j ź p i 1 ,j 1 qR i 1 p B 1 q X n i 1 ,j 1 ﬁ ﬂ ˇ ˇ ˇ ˇ ˇ ˇ “ O ˆ 1 n ˙ . (2.18) If any edge of the multigraph C Y C 1 zp B Y B 1 q has multiplicit y one, then the expectation in ( 2.18 ) v anishes. Otherwise, ev ery edge app ears with multiplicit y at least t wo, and the pro of of ( 2.18 ) is iden tical to that of ( 2.13 ). Consequen tly , E ˇ ˇ tr pp Y n q k q ˇ ˇ 2 “ E ˇ ˇ tr pp X n q k q ˇ ˇ 2 ` | tr pp E n q k q| 2 ` 2 E “ tr pp X n q k q tr pp E n q k q ‰ ` O ˆ 1 n ˙ . By Assumption 2 , | tr pp E n q k q| 2 is uniformly bounded, and E | tr pp X n q k q| 2 is bounded by Lemma 2.8 . This pro v es the claim. □ All the necessary ingredients are now in place to complete the pro of of Theorem 1.1 . Pr o of of The or em 1.1 . Recall the notation from Prop osition 2.4 . W e ﬁrst show that q D n p z q „ n b n p z q det ` I ´ z X n,D ˘ „ n b n p z q κ D p z q exp p´ F q , (2.19) where κ D p z q “ b 1 ´ z 2 E p A D 1 , 1 q 2 . Moreov er, in what follows set Q D n p z q “ b n p z q κ D p z q exp p´ F q , Q n p z q “ b n p z q κ p z q exp p´ F q . Notice that for z P C , the series ř 8 k “ 1 z k k p Y n,D q k is w ell-deﬁned for | z | small enough, and we can express q D n p z q as (2.20) q D n p z q “ exp ˜ ´ 8 ÿ k “ 1 tr ` p Y n,D q k ˘ z k k ¸ . By Prop osition 6.1 of [ Cos23 ], we can rewrite, for | z | small enough, exp ˜ ´ 8 ÿ k “ 1 tr ` p Y n,D q k ˘ z k k ¸ “ 1 ` n ÿ k “ 1 P k ´ tr p Y n,D q , . . . , tr ` p Y n,D q k ˘ ¯ z k k ! , 16 EXTREME EIGENV ALUES AND EIGENVECTORS for some p olynomials P k whic h do not dep end on n . By analytic contin uation, q D n p z q “ 1 ` n ÿ k “ 1 P k ´ tr p Y n,D q , . . . , tr ` p Y n,D q k ˘ ¯ z k k ! for any z P C . Th us, it suﬃces to examine the joint law of ` tr p Y n,D q , . . . , tr ` p Y n,D q k ˘˘ for any k P N . In this case, we com bine Prop osition 2.7 , Lemma 2.8 , and Lemma 2.9 to conclude (2.21) ` tr p Y n,D q , . . . , tr ` p Y n,D q k ˘˘ „ n ` tr p X n,D q , . . . , tr ` p X n,D q k ˘˘ ` ` tr p E n q , . . . , tr ` p E n q k ˘˘ „ n ` Z 1 ` mean D 1 , ? 2 Z 2 ` mean D 2 , . . . , ? k Z k ` mean D k ˘ ` ` tr p E n q , . . . , tr ` p E n q k ˘˘ . Notice that the Gaussian random v ariables Z k do not depend on D . By Prop osition 2.2 and ( 2.21 ), we deduce that ( 2.19 ) holds. W e con tin ue with the pro of of ( 1.4 ). Fix an integer m ą 0, and an m -tuple p z 1 , . . . , z m q P D p 0 , 1 q m . Let φ : R 2 m Ñ R b e a b ounded Lipschitz function. Since for all z P D p 0 , 1 q , lim D Ñ8 κ D p z q “ κ p z q , it follows that sup n ˇ ˇ E φ ` Q n p z 1 q , . . . , Q n p z m q ˘ ´ E φ ` Q D n p z 1 q , . . . , Q D n p z m q ˘ ˇ ˇ Ý Ý Ý Ý Ñ D Ñ8 0 . Therefore, ˇ ˇ E φ ` q n p z 1 q , . . . , q n p z m q ˘ ´ E φ ` Q n p z 1 q , . . . , Q n p z m q ˘ ˇ ˇ ď ˇ ˇ E φ ` q n p z 1 q , . . . , q n p z m q ˘ ´ E φ ` q D n p z 1 q , . . . , q D n p z m q ˘ ˇ ˇ ` ˇ ˇ E φ ` q D n p z 1 q , . . . , q D n p z m q ˘ ´ E φ ` Q D n p z 1 q , . . . , Q D n p z m q ˘ ˇ ˇ ` ˇ ˇ E φ ` Q D n p z 1 q , . . . , Q D n p z m q ˘ ´ E φ ` Q n p z 1 q , . . . , Q n p z m q ˘ ˇ ˇ . (2.22) The ﬁrst term on the right-hand side is b ounded b y a p ositive num ber ε D indep enden t of n and con v erging to zero as D Ñ 8 by Prop osition 2.4 . The second term con v erges to zero as n Ñ 8 since q D n p z q „ n Q D n p z q . W e just show ed that the third term can be controlled similarly to the ﬁrst te rm. Thus, the left-hand side conv erges to zero as n Ñ 8 . By applying Prop osition 2.2 , we obtain q n „ n Q n . Next we prov e ( 1.3 ). Set S n p z q “ b n p z q det ` I ´ z X n ˘ , S D n p z q “ b n p z q det ` I ´ z X n,D ˘ . Giv en the b ounds from ( 2.22 ) and ( 2.19 ), it is suﬃcient to pro v e that sup n ˇ ˇ E φ ` S n p z 1 q , . . . , S n p z m q ˘ ´ E φ ` S D n p z 1 q , . . . , S D n p z m q ˘ ˇ ˇ Ý Ý Ý Ý Ñ D Ñ8 0 . The latter can b e pro v en easily b y using Assumption 2 to b ound b n p z q and comparing det p I ´ z X n q with det p I ´ z X n,D q as is done in Lemma 3.3 of [ BCGZ22 ]. □ 3. Comp arison with a Ga ussian ma trix The goal of this section is to compare the spectral prop erties of matrix X n as deﬁned in ( 1.1 ) with analogous quan tities of a Gaussian random matrix G n P R n ˆ n with i.i.d. centered real Gaussian en tries, eac h with v ariance n ´ 1 . W e mostly rely on results from [ BvH24 ]. Let ξ P C b e such that | ξ | ą 1. Consider the following matrices (3.1) H n p ξ q “ « 0 X n ´ ξ I n p X n ´ ξ I n q ‹ 0 ﬀ and S n p ξ q “ « 0 G n ´ ξ I n p G n ´ ξ I n q ‹ 0 ﬀ . As is well kno wn, the set of eigenv alues of H n p ξ q counting multiplicities coincides with the union of the set of singular v alues of X n ´ ξ I n coun ting multiplicities and the set of the opp osites of these singular v alues. A similar remark holds for S n p ξ q and G n ´ ξ I n . F or simplicit y we will often write H n , S n instead of H n p ξ q , S n p ξ q . W e start with a comparison of the empirical sp ectral distribution of the matrices. EXTREME EIGENV ALUES AND EIGENVECTORS 17 Prop osition 3.1. The sp e ctr al me asur es of H n and S n ar e asymptotic al ly e quivalent, that is, writing ν H n “ 1 2 n ÿ i Pr 2 n s δ λ i p H n q and ν S n “ 1 2 n ÿ i Pr 2 n s δ λ i p S n q , it holds that ν H n „ ν S n as r andom variables value d in the sp ac e of pr ob ability me asur es on R . Pr o of. F ollows directly by the discusson in Subsection 10.3 of [ R T19 ]. □ Next we prov e a classical result for sub-gaussian random v ariables. Lemma 3.2. L et Assumption 5 hold. Then for some c onstant C 1 ą 0 , lim n Ñ8 P ˜ max i,j | X n i,j | ď C 1 ˆ log n K n ˙ 1 { 2 ¸ “ 1 . (3.2) Pr o of. By the union b ound and Assumption 5 we get P ˜ max i,j | X n i,j | ą C 1 ˆ log n K n ˙ 1 { 2 ¸ ď P ˜ max i,j | A n i,j | ? K n ą C 1 ˆ log n K n ˙ 1 { 2 ¸ “ P ˆ max i,j | A n i,j | ą C 1 p log n q 1 { 2 ˙ ď 2 n 2 exp p´ C p C 1 q 2 log p n qq . It remains to choose C 1 large enough to conclude. □ F or the empirical spectral distribution the ﬁniteness of the second moment of the en tries of A n w as suﬃcient. F or ﬁner results one needs to make more assumptions for the matrix A n . Recall that C ` “ t z P C : ℑ z ą 0 u and that s n p M q denotes the least singular v alue of any n ˆ n matrix M . W e no w present our comparison result, a corollary of [ BvH24 ]. Theorem 3.3. L et Assumptions 1 and 5 hold. L et G n b e a n ˆ n matrix with i.i.d. c enter e d r e al Gaussian entries e ach with varianc e n ´ 1 and the matric es H n p ξ q and S n p ξ q b e deﬁne d by ( 3.1 ) . L et ξ P C with | ξ | ą 1 . Assume that lim n Ñ8 log 9 p n q K n “ 0 , Then (a) for every ε ą 0 , lim n Ñ8 P p| s n p X n ´ ξ I q ´ s n p G n ´ ξ I q| ě ε q “ 0 , (b) for every z P C ` , lim n Ñ8 › › E p S n ´ z I q ´ 1 ´ E p H n ´ z I q ´ 1 › › “ 0 . R emark 3.4 . In the previous theorem, it turns out that assumption log 9 p n q{ K n Ñ 0 is required to prov e item (a) . The lighter assumption log p n q{ K n Ñ 0 is suﬃcient to establish item (b) , see the pro of b elo w. Pr o of. As a consequence of Assumption 5 , the following estimate holds (3.3) E max i,j Pr n s | A n ij | 2 ď C 1 log n , for some constant C ą 0 (see, for instance, [ V er18 , Exercises 2.26 and 2.44]). Moreov er, the singular v alue s n p G n ´ ξ I q is p ositive with probability one and coincides on this probability one set with the smallest p ositive eigen v alue of S n . In order to establish (a) we shall rely on [ BvH24 , Theorem 2.8]. Notice ﬁrst that | s n p X n ´ ξ I q ´ s n p G n ´ ξ I q| ď d H p σ p H n q , σ p S n qq , where σ p H n q and σ p S n q are resp ectively the sp ectra of H n and S n . The following quantities whose estimates are straightforw ard app ear in the statemen t of [ BvH24 , Theorem 2.8]: κ “ › › E p H ´ E H q 2 › › 2 “ 1 , κ ˚ “ sup } v } , } w }“ 1 ´ E |x v , p H ´ E H q w y| 2 ¯ 1 { 2 ď 2 ? n , R “ ˆ E max ij | X n ij | 2 ˙ 1 { 2 ď ˆ C 1 log p n q K n ˙ 1 { 2 . 18 EXTREME EIGENV ALUES AND EIGENVECTORS No w the theorem states that: P " | s n p G n ´ ξ I q ´ s n p X n ´ ξ I q| ě C 0 ε p t q ; max i,j | X ij | ď R * ď 2 n exp p´ t q , for every t ě 0 with the conditions: p i q R ě a κ R ` ? 2 R and p ii q ε R p t q “ κ ˚ ? t ` R 1 { 3 κ 2 { 3 t 2 { 3 ` R t , and where C 0 is a univ ersal constant. W e ﬁrst set R “ C 2 ´ log p n q K n ¯ 1 { 4 and notice that for this v alue, P t max i,j | X ij | ď R u Ñ n 1. In fact, using estimate ( 3.3 ) w e hav e P " max i,j | X ij | ą R * ď C 1 C 2 2 d log p n q K n Ý Ý Ý Ñ n Ñ8 0 , b y assumption. Now setting t “ C 3 ´ K n log p n q ¯ 1 { 8 , we get ε R p t q “ C 1 { 3 2 C 2 { 3 3 ` o p 1 q . Notice that with such a choice, 2 ne ´ t “ 2 exp # log p n q ´ C 3 ˆ K n log p n q ˙ 1 { 8 + “ 2 exp # log p n q « 1 ´ C 3 ˆ K n log 9 p n q ˙ 1 { 8 ﬀ+ Ý Ý Ý Ñ n Ñ8 0 b y the condition K n { log 9 p n q Ñ 8 . It remains to choose C 3 so that C 1 { 3 2 C 2 { 3 3 “ ε to conclude that P t| s n p G n ´ ξ I q ´ s n p X n ´ ξ I q| ě 2 C 0 ε u ď P " | s n p G n ´ ξ I q ´ s n p X n ´ ξ I q| ě C 0 ε p t q ; max i,j | X ij | ď R * ` P " max i,j | X ij | ą R * Ý Ý Ý Ñ n Ñ8 0 . In order to establish (b) we shall rely on [ BvH24 , Theorem 2.11] which yields that for every z P C ` › › E p z I ´ H q ´ 1 ´ E p z I ´ S q ´ 1 › › ď κ ˚ ` R 1 { 10 ℑ 2 p z q “ 1 ℑ 2 p z q ˜ 2 ? n ` ˆ log p n q K n ˙ 1 { 20 ¸ . Pro of of Theorem 3.3 is completed. □ R emark 3.5 . The results of [ BvH24 ] are fairly general. One may relax the sub-Gaussian assumption (Assumption 5 ) at the cost of increasing the sparsity parameter K n and still ha ve an analogue of Theorem 3.3 . W e no w prov e a concentration result. Recall that X n ’s entries write X n ij “ B n ij A n ij ? K n . Lemma 3.6. Assume that E | A n 11 | 8 ă 8 . L et z P C ` , ε ą 0 and c onsider two se quenc es p ˜ w 2 n q and p ˜ q 2 n q of unit ve ctors in C 2 n , wher e ˜ w 2 n i “ ˜ q 2 n i “ 0 for i P t n ` 1 , ¨ ¨ ¨ , 2 n u . Then lim n Ñ8 P ` ˇ ˇ xp H n p ξ q ´ z I q ´ 1 ˜ w 2 n , ˜ q 2 n y ´ E xp H n p ξ q ´ z I q ´ 1 ˜ w 2 n , ˜ q 2 n y ˇ ˇ ě ϵ ˘ “ 0 . R emark 3.7 . In the pro of b elow, the condition E | A 11 | 4 ă 8 app ears in estimating the v ariance of a quadratic form, see for instance ( 3.5 ). The eigh t moment is required when relying on [ HLNV13 , Theorem 3.6]. Pr o of. W e write ˜ w 2 n “ ˜ w n 0 n ¸ and ˜ q 2 n “ ˜ q n 0 n ¸ , where w n , q n P C n and 0 n is the null vector in C n . W e will so on drop the index n and simply write w , q , I instead of w n , q n , I n . In the sequel, C denotes a constant whose v alue may c hange from line to line. By the Sc hur complement formula, w e hav e xp H n p ξ q ´ z I 2 n q ´ 1 ˜ w 2 n , ˜ q 2 n y “ z x w n , ` ´ z 2 I n ` p X ´ ξ I n qp X ´ ξ I n q ‹ ˘ ´ 1 q n y , EXTREME EIGENV ALUES AND EIGENVECTORS 19 and we are led to study the concentration of the quadratic form x w , Q q y with Q “ z ` ´ z 2 I ` p X ´ ξ I qp X ´ ξ I q ‹ ˘ ´ 1 . Notice that Q b eing the top-left corner of matrix p H n ´ z I q ´ 1 , w e immediately get } Q } ď p ℑ p z qq ´ 1 . Denote b y Y “ X ´ ξ I and let the p y i q ’s b eing the columns of matrix Y . In particular, y i “ x i ´ ξ e i and Q “ z ` ´ z 2 ` Y Y ‹ ˘ ´ 1 “ z ˜ ´ z 2 ` n ÿ k “ 1 y k y ‹ k ¸ ´ 1 . F or further use, w e introduce Q i “ z ` ´ z 2 ` ř k ‰ i y k y ‹ k ˘ ´ 1 . Denote by f p y 1 , ¨ ¨ ¨ , y n q “ x w , Q q y . Let ˇ f i b e the function f ev aluated at p y 1 , ¨ ¨ ¨ , y i ´ 1 , ˇ y i , y i ` 1 , ¨ ¨ ¨ , y n q where ˇ y i is an indep endent copy of y i . By Efron-Stein’s inequality [ BLB03 , Theorem 3.1] we hav e v ar p f q ď 1 2 n ÿ i “ 1 E | f ´ ˇ f i | 2 . W e will rely on the follo wing elementary facts. Let M P C n ˆ n a deterministic matrix, then E p y ‹ i M y i q “ 1 n T race p M q ` | ξ | 2 M ii , (3.4) v ar p y ‹ i M y i q ď C ˆ E | A 11 | 4 nK n T race p M M ‹ q ` | ξ | 2 p M M ‹ q ii n ˙ . (3.5) The function z ÞÑ y ‹ i Q i p z q y i is the Stieltjes transform of a non-negativ e measure, and the function z ÞÑ ´ 1 z ` y ‹ i Q i p z q y i . is the Stieltjes transform of a probabilit y measure. In particular ˇ ˇ ˇ ˇ 1 z ` y ‹ i Q i p z q y i ˇ ˇ ˇ ˇ ď 1 ℑ p z q . In the sequel, we denote by E i “ E p ¨ | y k , k ‰ i q and b y v ar i the associated conditional v ariance. Using Sherman-Morrisson’s inequality , we get E ˇ ˇ f ´ ˇ f i ˇ ˇ 2 “ EE i ˇ ˇ ˇ ˇ y ‹ i Q i q w ‹ Q i y i z ` y ‹ i Q i y i ´ ˇ y i ‹ Q i q w ‹ Q i ˇ y i z ` ˇ y ‹ i Q i y i ˇ ˇ ˇ ˇ 2 , p a q ď 2 EE i ˇ ˇ ˇ ˇ ˇ y ‹ i Q i q w ‹ Q i y i z ` y ‹ i Q i y i ´ E i ` y ‹ i Q i q w ‹ Q i y i ˘ z ` E i p y ‹ i Q i y i q ˇ ˇ ˇ ˇ ˇ 2 ` 2 EE i ˇ ˇ ˇ ˇ ˇ ˇ y i ‹ Q i q w ‹ Q i ˇ y i z ` ˇ y ‹ i Q i ˇ y i ´ E i ` ˇ y ‹ i Q i q w ‹ Q i ˇ y i ˘ z ` E i p ˇ y ‹ i Q i ˇ y i q ˇ ˇ ˇ ˇ ˇ 2 , “ 4 EE i ˇ ˇ ˇ ˇ ˇ y ‹ i Q i q w ‹ Q i y i z ` y ‹ i Q i y i ´ E i ` y ‹ i Q i q w ‹ Q i y i ˘ z ` E i p y ‹ i Q i y i q ˇ ˇ ˇ ˇ ˇ 2 , where p a q follo ws from the introduction of the auxiliary term E i ` y ‹ i Q i q w ‹ Q i y i ˘ z ` E i p y ‹ i Q i y i q “ E i ` ˇ y ‹ i Q i q w ‹ Q i ˇ y i ˘ z ` E i p ˇ y ‹ i Q i ˇ y i q , 20 EXTREME EIGENV ALUES AND EIGENVECTORS and the elementary inequality | a ` b | 2 ď 2 | a | 2 ` 2 | b | 2 . Introducing appropriate auxiliary terms and pro ceeding similarly , w e get E ˇ ˇ f ´ ˇ f i ˇ ˇ 2 ď 8 EE i ˇ ˇ ˇ ˇ ˇ y ‹ i Q i q w ‹ Q i y i z ` y ‹ i Q i y i ´ E i ` y ‹ i Q i q w ‹ Q i y i ˘ z ` y ‹ i Q i y i ˇ ˇ ˇ ˇ ˇ 2 ` 8 EE i ˇ ˇ ˇ ˇ E i ` y ‹ i Q i q w ‹ Q i y i ˘ " 1 z ` y ‹ i Q i y i ´ 1 z ` E i p y ‹ i Q i y i q * ˇ ˇ ˇ ˇ 2 , ď 8 ℑ 2 p z q E v ar i p y ‹ i Q i q w ‹ Q i y i q ` 8 ℑ 4 p z q E ˇ ˇ E i p y ‹ i Q i q w ‹ Q i y i q ` y ‹ i Q i y i ´ E i p y ‹ i Q i y i q ˘ ˇ ˇ 2 , “ O z ` E v ar i p y ‹ i Q i q w ‹ Q i y i q ˘ ` O z ´ E ˇ ˇ E i p y ‹ i Q i q w ‹ Q i y i q ` y ‹ i Q i y i ´ E i p y ‹ i Q i y i q ˘ ˇ ˇ 2 ¯ . (3.6) W e ﬁrst estimate v ar i p y ‹ i Q i q w Q i y i q . By ( 3.5 ) w e hav e v ar i p y ‹ i Q i q w ‹ Q i y i q ď C ˜ T race p Q i q w ‹ Q i r Q i s ‹ w q ‹ r Q i s ‹ q nK n ` | ξ | 2 ` Q i q w ‹ Q i r Q i s ‹ w q ‹ r Q i s ‹ ˘ ii n ¸ , “ O z ˆ 1 nK n ˙ ` O z ,ξ ˜ ` Q i q q ‹ r Q i s ‹ ˘ ii n ¸ . (3.7) W e no w estimate E i p y ‹ i Q i q w ‹ Q i y i q . By ( 3.4 ) w e hav e E i p y ‹ i Q i q w ‹ Q i y i q “ 1 n T race p Q i q w ‹ Q i q ` | ξ | 2 ` Q i q w ‹ Q i ˘ ii “ O z ˆ 1 n ˙ ` | ξ | 2 ` Q i q w ‹ Q i ˘ ii . Notice that ˇ ˇ ` Q i q w ‹ Q i ˘ ii ˇ ˇ 2 “ ` Q i q w ‹ Q i ˘ ii ˆ ` r Q i s ‹ w q ‹ r Q i s ‹ ˘ ii ď ` Q i q w ‹ Q i r Q i s ‹ w q ‹ r Q i s ‹ ˘ ii ď 1 ℑ 2 p z q ` Q i q q ‹ r Q i s ‹ ˘ ii . Hence (3.8) ˇ ˇ E i p y ‹ i Q i q w ‹ Q i y i q ˇ ˇ 2 “ O z ˆ 1 n 2 ˙ ` O z ,ξ ´ ˇ ˇ ` Q i q w ‹ Q i ˘ ii ˇ ˇ 2 ¯ “ O z ˆ 1 n 2 ˙ ` O z ,ξ `` Q i q q ‹ r Q i s ‹ ˘ ii ˘ . W e ﬁnally estimate v ar i p y ‹ i Q i y i q . By ( 3.5 ) w e hav e (3.9) v ar i p y ‹ i Q i y i q ď r K ˜ T race p Q i r Q i s ‹ q nK n ` | ξ | 2 “ Q i r Q i s ‹ ‰ ii n ¸ “ O z ˆ 1 K n ˙ ` O z ,ξ ˆ 1 n ˙ “ O z ,ξ ˆ 1 K n ˙ . Notice that the ﬁnal upp er estimate of v ar i p y ‹ i Q i y i q ab ov e is deterministic. Noticing that E i ␣ | E i U | 2 ˆ | V | 2 ( “ | E i U | 2 E i | V | 2 , and using ( 3.8 )-( 3.9 ), w e get EE i ˇ ˇ E i p y ‹ i Q i q w ‹ Q i y i q ` y ‹ i Q i y i ´ E i p y ‹ i Q i y i q ˘ ˇ ˇ 2 “ E ! ˇ ˇ E i p y ‹ i Q i q w ‹ Q i y i q ˇ ˇ 2 E i ˇ ˇ y ‹ i Q i y i ´ E i p y ‹ i Q i y i q ˇ ˇ 2 ) , “ E ! ˇ ˇ E i p y ‹ i Q i q w ‹ Q i y i q ˇ ˇ 2 v ar i p y ‹ i Q i y i q ) , “ O z ,ξ ˆ 1 n 2 K n ˙ ` O z ,ξ ˜ E ` Q i q q ‹ r Q i s ‹ ˘ ii K n ¸ . (3.10) Plugging back estimates ( 3.7 ) and ( 3.10 ) into ( 3.6 ) and summing ov er i ﬁnally yields v ar p f q ď 1 2 ÿ i E | f ´ ˇ f i | 2 “ O ξ,z ˆ 1 K n ` ř i E p Q i q q ‹ r Q i s ‹ q ii K n ˙ . It remains to notice that ÿ i E ` Q i q q ‹ r Q i s ‹ ˘ ii “ O z p 1 q b y [ HLNV13 , Theorem 3.6] to conclude. □ W e no w present fairly standard results concerning the Gaussian matrix S n . EXTREME EIGENV ALUES AND EIGENVECTORS 21 Theorem 3.8. L et ξ P C with | ξ | ą 1 . L et G n b e a n ˆ n matrix with i.i.d. r e al Gaussian entries e ach with varianc e n ´ 1 and S n p ξ q the 2 n ˆ 2 n matrix deﬁne d by ( 3.1 ) . The fol lowing facts hold true for matrix S n p ξ q . (a) Ther e exists a pr ob ability me asur e µ ξ such that 1 2 n ÿ i Pr 2 n s δ λ i p S n q ù ñ n Ñ8 µ ξ a.s. (b) The pr ob ability me asur e µ ξ is symmetric and has a density supp orte d in p´ C ξ , ´ c ξ q Y p c ξ , C ξ q for some p ositive c onstants 0 ă c ξ ă C ξ . (c) If m ξ denotes the Stieltjes tr ansform of µ ξ , then m ξ is the unique function that satisﬁes the fol lowing ﬁxe d p oint e quation ´ 1 m ξ p w q “ w ` m ξ p w q ´ | ξ | 2 w ` m ξ p w q , with ℑ p m ξ p w qq , ℑ p w q ą 0 , (3.11) (d) Ther e exists a p ositive c onstant ˇ c ξ such that lim n Ñ8 P p s n p G n ´ ξ I q ě ˇ c ξ q “ 1 . L et ˜ w 2 n , ˜ q 2 n b e two deterministic unit ve ctors in C 2 n satisfying ˜ w 2 n i “ ˜ q 2 n i “ 0 for i ě n ` 1 . (e) L et η P R ` , then lim n Ñ8 ˇ ˇ @ ˜ w 2 n , p S n ´ iη I q ´ 1 ˜ q 2 n D ´ m ξ p iη qx ˜ w 2 n , ˜ q 2 n y ˇ ˇ “ 0 a.s. . (f ) L et z P C ` , then for every ε ą 0 , lim n Ñ8 P ` ˇ ˇ @ ˜ w 2 n , p S n ´ z I q ´ 1 ˜ q 2 n D ´ @ ˜ w 2 n , E p S n ´ z I q ´ 1 ˜ q 2 n D ˇ ˇ ě ε ˘ “ 0 . Pr o of. Random matrix models lik e S n are v ery popular and ha v e been hea vily studied. (a) and (b) can be found in Proposition 3.1 of [ BYY14 ]; (c) can b e found in [ CESX23 , (2.17)]; (d) can b e pro ven by a direct application of [ DS07 , Theorem 1.1]. Finally (e) and (f ) are consequences of [ HLNV13 , Theorem 1.1]. □ Corollary 3.9. L et η ą 0 and m ξ the Stieltjes tr ansform deﬁne d in The or em 3.8 - (c) , then one has: lim η Ñ 0 ℑ p m ξ p iη qq η “ 1 | ξ | 2 ´ 1 Pr o of. Let ρ ξ denote the density of µ ξ , notice that ρ ξ is symmetric. Recall that µ ξ is supp orted in p´ C ξ , ´ c ξ q Y p c ξ , C ξ q for p ositiv e constants c ξ , C ξ . First, deﬁne the function h p η q : “ ℑ p m ξ p iη qq η “ 2 ż C ξ c ξ ρ ξ p x q x 2 ` η 2 dx . Then h p η q is Lipsc hitz contin uous on a small in terv al p 0 , ε q with ε ă c ξ since | h p η 1 q ´ h p η 2 q| ď 4 | η 1 ´ η 2 | ε ż C ξ c ξ ρ ξ p x q p x 2 ` η 2 1 qp x 2 ` η 2 2 q dx ď 4 ε c 4 ξ | η 1 ´ η 2 | . In particular, the limit lim η Ñ 0 h p η q exists. The symmetry of the density ρ ξ yields that m ξ p iη q “ ´ m ξ p iη q hence ℜ m ξ p iη q “ 0 . Rewriting the ﬁxed p oint equation in Theorem 3.8 - (c) in terms of function h p η q yields 1 “ h p η q η 2 p 1 ` h p η qq ` | ξ | 2 h p η q 1 ` h p η q . (3.12) T aking the limit of ( 3.12 ) as η Ñ 0 we end up with the desired result: h p 0 q “ 1 | ξ | 2 ´ 1 . □ W e are no w in p osition to compare quadratic forms based on the resolven t of H n and on the resolven t of S n . 22 EXTREME EIGENV ALUES AND EIGENVECTORS Corollary 3.10. L et A n satisfy Assumption ( 5 ) , z P C ` , ε ą 0 and lim n Ñ8 log n K n “ 0 . L et ˜ w 2 n , ˜ q 2 n P C 2 n b e deterministic unit ve ctors satisfying ˜ w 2 n i “ ˜ q 2 n i “ 0 for i ě n ` 1 , then lim n Ñ8 P ` ˇ ˇ xp H n p ξ q ´ z I q ´ 1 ˜ w 2 n , ˜ q 2 n y ´ xp S n p ξ q ´ z I q ´ 1 ˜ w 2 n , ˜ q 2 n y ˇ ˇ ě ϵ ˘ “ 0 . Pr o of. In the notations b elow, we drop the indices. The claim follows from the inequality ˇ ˇ xp H ´ z I q ´ 1 ˜ w , ˜ q y ´ xp S ´ z I q ´ 1 ˜ w , ˜ q y ˇ ˇ ď ˇ ˇ xp H ´ z I q ´ 1 ˜ w , ˜ q y ´ E xp H ´ z I q ´ 1 ˜ w , ˜ q y ˇ ˇ ` ˇ ˇ xp S ´ z I q ´ 1 ˜ w , ˜ q y ´ E xp S ´ z I q ´ 1 ˜ w , ˜ q y ˇ ˇ ` › › E p S ´ z I q ´ 1 ´ E p H ´ z I q ´ 1 › › . The ﬁrst term of the r.h.s. go es to zero in probabilit y by Lemma 3.6 ; the second term go es to zero by Theorem 3.8 - (f ) ; the last term goes to zero by Theorem 3.3 - (b) . □ 4. Proof of Theorem 1.6 Recall the deﬁnition of X n in ( 1.1 ) and the fact that Y n “ X n ` u n p v n q ‹ . In all this section, we shall assume without generality loss that x v n , u n y Ý Ý Ý Ñ n Ñ8 ξ P C with | ξ | ą 1 , since it is suﬃcient to establish the conv ergence in probabilit y to all sub-sequential limits of x v n , u n y . W e start our analysis with a w ell-kno wn linear algebra result (see, e.g. , [ BGN11 , T ao13 ]) that we pro ve for completeness. Lemma 4.1. L et z 0 R σ p X n q . Then, z 0 P σ p Y n q if and only if 1 ` @ p X n ´ z 0 I q ´ 1 u n , v n D “ 0 . The c ase b eing, a right eigenve ctor c orr esp onding to the eigenvalue z 0 of Y n is p X n ´ z 0 I q ´ 1 u n . Pr o of. F or the ﬁrst part, since z 0 is not an eigen v alue of X n and b y the prop e rt y that det p I ` AB q “ det p I ` B A q for rectangular matrices A and B with compatible dimensions, w e get that det p Y n ´ z 0 I q det p X n ´ z 0 I q “ det p I ` p X n ´ z 0 I q ´ 1 u n p v n q ‹ q “ 1 ` @ p X n ´ z 0 I q ´ 1 u n , v n D . The claim follo ws. F or the second part, for z 0 R σ p X n q , we hav e that p Y n ´ z 0 I qp X n ´ z 0 I q ´ 1 u n “ u n ` u n p v n q ‹ p X n ´ z 0 I q ´ 1 u n “ `@ p X n ´ z 0 I q ´ 1 u n , v n D ` 1 ˘ u n . Due to the ﬁrst part of the lemma, if z 0 is an eigenv alue of Y n , the righ t hand side of this expression is zero. Th us Y n p X n ´ z 0 I q ´ 1 u n “ z 0 p X n ´ z 0 I q ´ 1 u n whic h is the required result. □ Let us brieﬂy presen t the strategy of proof. Thanks to the former result, we are led to study the b ehavior of B u n } u n } , p X n ´ λ max p Y n q I q ´ 1 u n }p X n ´ λ max p Y n q I q ´ 1 u n } F on an appropriate probabilit y ev ent. With the help of the results of the former section, we ﬁrst sho w that p X n ´ λ max p Y n q I q ´ 1 can b e replaced with p X n ´ ξ I q ´ 1 in this expression. This is the aim of Lemma 4.2 b elow. With the help of Theorem 1.1 , we then consider the asymptotics of x u n , p X n ´ ξ I q ´ 1 u n y{} u n } 2 (Lemma 4.3 ). The remainder of the proof consists in studying }p X n ´ ξ I q ´ 1 u n }{} u n } with help of the results of Section 3 again. EXTREME EIGENV ALUES AND EIGENVECTORS 23 Lemma 4.2. Ther e exists a se quenc e p E 4 . 2 n q of pr ob ability events such that 1 E 4 . 2 n Ñ 1 in pr ob ability, the smal lest singular values of X n ´ ξ I and X n ´ λ max p Y n q I ar e lower b ounde d by p ositive c onstants on E 4 . 2 n , and mor e over, it holds that 1 E 4 . 2 n }p X n ´ λ max p Y n q I q ´ 1 ´ p X n ´ ξ I q ´ 1 } P Ý Ý Ý Ñ n Ñ8 0 . Pr o of. W e mainly need to control the smallest singular v alue s n p X n ´ ξ I q , and to use Corollary 1.5 , whic h sho ws in our context that λ max p Y n q P Ý Ý Ý Ñ n Ñ8 ξ . T o control s n p X n ´ ξ I q , we apply Theorems 3.3 - (a) and 3.8 - (d) to obtain the existence of a constant c ą 0 satisfying lim n P t s n p X n ´ ξ I q ě c u “ 1 . Deﬁning the ev ent E 4 . 2 n “ t s n p X n ´ ξ I q ě c u X t| λ max p Y n q ´ ξ | ď c { 2 u , w e know from what precedes that P t E 4 . 2 n u Ñ n 1. Moreov er, by W eyl’s inequalit y , w e obtain that s n p X n ´ λ max p Y n q I q ě s n p X n ´ ξ I q ´ | ξ ´ λ max p Y n q| . Therefore, s n p X n ´ λ max p Y n q I q ě c { 2 on E 4 . 2 n , and b oth matrices X n ´ ξ I and X n ´ λ max p Y n q I hav e their smallest singular v alues lo w er b ounded by a p ositive constan t on E 4 . 2 n . In particular, the expression 1 E 4 . 2 n }p X n ´ λ max p Y n q I q ´ 1 ´ p X n ´ ξ I q ´ 1 } is well-deﬁned. On E 4 . 2 n , we furthermore hav e }p X n ´ λ max p Y n q I q ´ 1 ´ p X n ´ ξ I q ´ 1 } “ }p X n ´ λ max p Y n q I q ´ 1 p X n ´ ξ I q ´ 1 p λ max p Y n q ´ ξ q} , ď | λ max p Y n q ´ ξ | }p X n ´ λ max p Y n q I q ´ 1 } }p X n ´ ξ I q ´ 1 } , ď 2 c 2 | λ max p Y n q ´ ξ | , and the second statement follows from the conv ergence of λ max p Y n q to ξ in probability . □ Next we turn our attention to xp X n ´ ξ I q ´ 1 u n , u n y on the even t where p X n ´ ξ I q is inv ertible. Lemma 4.3. L et E 4 . 3 n b e the event wher e p X n ´ ξ I q is invertible. Then, 1 E 4 . 3 n Ñ n 1 in pr ob ability, and 1 E 4 . 3 n 1 } u n } 2 @ p X n ´ ξ I q ´ 1 u n , u n D P Ý Ý Ý Ñ n Ñ8 ´ 1 ξ . Pr o of. The conv ergence 1 E 4 . 3 n Ñ n 1 in probability follo ws obviously from, e.g. , Theorem 1.4 . Arguing as in the pro of of Lemma 4.1 , w e furthermore hav e 1 E 4 . 3 n det p I ´ ξ ´ 1 p X n ` u n p u n q ‹ qq “ 1 E 4 . 3 n ` 1 ` @ u n , p X n ´ ξ I q ´ 1 u n D˘ det p I ´ ξ ´ 1 X n q . By Assumptions 2 and 4 , the sequence p} u n }q con v erges to a limit β ą 0 along a subsequence that we still denote as p n q . W e ﬁx this subsequence. Setting ˇ E n “ u n p u n q ‹ and ˇ Y n “ X n ` ˇ E n , and deﬁning the H 2 –v alued random vector r q ˇ Y n q X n s T as ˜ q ˇ Y n p z q q X n p z q ¸ “ ˜ det p I ´ z ˇ Y n q det p I ´ z X n q ¸ , w e easily see that the sequence pr q ˇ Y n q X n s T q is tight in the space H 2 equipp ed with the pro duct distance, and furthermore, by insp ecting again the pro of of Theorem 1.1 (in particular, Prop osition 2.7 with ˇ E n “ u n p u n q ‹ and Lemma 2.8 ), that ˜ q ˇ Y n q X n ¸ law Ý Ý Ý Ñ n Ñ8 κ exp p´ F q ˜ b 8 1 ¸ with b 8 p z q “ 1 ´ β 2 z . 24 EXTREME EIGENV ALUES AND EIGENVECTORS By Slutsky’s theorem, we then get that 1 E 4 . 3 n ˜ q ˇ Y n q X n ¸ law Ý Ý Ý Ñ n Ñ8 κ exp p´ F q ˜ b 8 1 ¸ . By Skorokhod’s represen tation theorem, there exists a sequence of C 2 –v alued random v ariables pr p ˇ Y n p X n s T q and a C 2 –v alued random v ariable pr p ˇ Y 8 p X 8 s T q on a probability space r Ω, such that ˜ p ˇ Y n p X n ¸ law “ 1 E 4 . 3 n ˜ q ˇ Y n p 1 { ξ q q X n p 1 { ξ q ¸ and ˜ p ˇ Y 8 p X 8 ¸ law “ κ p 1 { ξ q exp p´ F p 1 { ξ qq ˜ b 8 p 1 { ξ q 1 ¸ , and pr p Y n p X n s T q con v erges to pr p ˇ Y 8 p X 8 s T q for all ˜ ω P r Ω. Recalling that κ exp p´ F q ‰ 0, it holds that the random v ariable p ˇ Y n { p X n con verges p oin twise to p ˇ Y 8 { p X 8 law “ b 8 p 1 { ξ q . This implies that 1 E 4 . 3 n ` 1 ` @ u n , p X n ´ ξ I q ´ 1 u n D˘ “ 1 E 4 . 3 n q ˇ Y n p 1 { ξ q q X n p 1 { ξ q P Ý Ý Ý Ñ n Ñ8 b 8 p 1 { ξ q “ 1 ´ β 2 ξ , and the result of the lemma follo ws. □ It remains to establish an asymptotic result for 1 } u n } 2 }p X n ´ ξ I q ´ 1 u n } 2 . It will b e more conv enien t to w ork with the hermitisation H n p ξ q of X n deﬁned in ( 3.1 ). F urthermore, it will also b e conv enient to introduce a small parameter η ą 0 and work on the resolv ent p H n ´ z I q ´ 1 of H n ev aluated at z “ iη . Sp eciﬁcally: Lemma 4.4. Ther e exists a se quenc e of events p E 4 . 4 n q such that H n is invertible on E 4 . 4 n , E 4 . 2 n Ă E 4 . 4 n , and 1 E 4 . 4 n }p H n q ´ 1 ´ p H n ´ iη I q ´ 1 } ď C 4 . 4 η for some c onstant C 4 . 4 ą 0 . Pr o of. Recall that λ is an eigen v alue of the Hermitian matrix H n if and only if λ or ´ λ is a singular v alue of X n ´ ξ I . Th us, the ev en t E 4 . 4 n “ t s n p X n ´ ξ I q ě c u where c ą 0 is the one c hosen in the pro of of Lemma 4.2 satisﬁes the ﬁrst tw o assertions of the statement. On the ev ent E 4 . 4 n , it holds that }p H n q ´ 1 } ď 1 { c . On the same even t, since the singular v alues of p H n ´ iη I q ´ 1 are of the form 1 {| λ k ´ iη | where the λ k ’s are the real eigen v alues of H n , we obtain that }p H n ´ iη I q ´ 1 } ď 1 { c . By the resolv ent identit y , on this ev en t, we therefore obtain the following estimate: }p H n q ´ 1 ´ p H n ´ iη I q ´ 1 } “ }p H n q ´ 1 p H n ´ iη I q ´ 1 η } ď }p H n q ´ 1 } ˆ }p H n ´ iη I q ´ 1 } η ď η c 2 . □ F or the resolven t p H n ´ iη I q ´ 1 w e hav e that Lemma 4.5. Consider a se quenc e of deterministic unit ve ctors ˜ w 2 n P C 2 n satisfying ˜ w 2 n i “ 0 for i P t n ` 1 , ¨ ¨ ¨ , 2 n u , then the fol lowing limit holds: }p H n ´ iη I q ´ 1 ˜ w 2 n } 2 P Ý Ý Ý Ñ n Ñ8 ℑ p m ξ p iη qq η , wher e m ξ is the Stieltjes tr ansform of the pr ob ability me asur e ξ ξ deﬁne d in the statement of The or em 3.8 . Pr o of. Denoting as ℑ M “ p M ´ M ‹ q{p 2 i q the imaginary part of a complex matrix, it holds by the resolven t’s iden tity that ` p H n ´ iη I q ´ 1 ˘ ‹ p H n ´ iη I q ´ 1 “ 1 η ℑ ` p H n ´ iη I q ´ 1 ˘ . F rom this, we conclude that }p H n ´ iη I q ´ 1 ˜ w 2 n } 2 “ @ p H n ´ iη I q ´ 1 ˜ w 2 n , p H n ´ iη I q ´ 1 ˜ w 2 n D “ A ` p H n ´ iη I q ´ 1 ˘ ‹ p H n ´ iη I q ´ 1 ˜ w 2 n , ˜ w 2 n E “ B 1 η ℑ ` p H n ´ iη I q ´ 1 ˘ w 2 n , w 2 n F , and the claim follows by combining Corollary 3.10 with Theorem 3.8 - (e) . □ EXTREME EIGENV ALUES AND EIGENVECTORS 25 W e are now ready to examine the asymptotic b ehavior of 1 } u n } 2 }p X n ´ ξ I q ´ 1 u n } 2 . Lemma 4.6. L et u n P C n b e a deterministic ve ctor, then the fol lowing limit holds: 1 E 4 . 4 n }p X n ´ ξ I q ´ 1 u n } } u n } P Ý Ý Ý Ñ n Ñ8 1 a | ξ | 2 ´ 1 . Pr o of. Denote by q 2 n P C 2 n the deterministic unit vector deﬁned by q 2 n “ ˜ u n {} u n } 0 n ¸ . Recall that H n is inv ertible on E 4 . 4 n and notice that on this even t p H n q ´ 1 writes p H n q ´ 1 “ ˜ 0 p X n ´ ξ I q ´‹ p X n ´ ξ I q ´ 1 0 ¸ . In particular 1 E 4 . 4 n }p X n ´ ξ I q ´ 1 u n } } u n } “ 1 E 4 . 4 n }p H n q ´ 1 q 2 n } . Fix an arbitrarily small ε ą 0 and choose η ą 0 small enough so that C 4 . 4 η ď ε 2 and d ℑ m ξ p η q η ą 1 a | ξ | 2 ´ 1 ´ ε 4 , whic h is p ossible by Corollary 3.9 . With this choice, w e hav e by Lemma 4.4 ˇ ˇ 1 E 4 . 4 n › › p H n q ´ 1 q 2 n › › ´ 1 E 4 . 4 n › › p H n ´ iη I q ´ 1 q 2 n › › ˇ ˇ ď 1 E 4 . 4 n › › p H n q ´ 1 q 2 n ´ p H n ´ iη I q ´ 1 q 2 n › › ď ε 2 . No w # ˇ ˇ ˇ ˇ ˇ 1 E 4 . 4 n }p X n ´ ξ I q ´ 1 u n } } u n } ´ 1 a | ξ | 2 ´ 1 ˇ ˇ ˇ ˇ ˇ ě ε + Ă # ˇ ˇ ˇ ˇ ˇ 1 E 4 . 4 n › › p H n ´ iη I q ´ 1 q 2 n › › ´ 1 a | ξ | 2 ´ 1 ˇ ˇ ˇ ˇ ˇ ě ε 2 + Ă # ˇ ˇ ˇ ˇ ˇ › › p H n ´ iη I q ´ 1 q 2 n › › ´ d ℑ m ξ p η q η ˇ ˇ ˇ ˇ ˇ ě ε 4 + . T aking the probability of b oth even ts, com bined with Lemma 4.5 , yields the desired result. □ W e conclude with the pro of of Theorem 1.6 . Pr o of of The or em 1.6 . W e need to show that ˇ ˇ ˇ ˇ B u n } u n } , ˜ u n F ˇ ˇ ˇ ˇ 2 P Ý Ý Ý Ñ n Ñ8 1 ´ 1 | ξ | 2 . T o this end, w e are allo w ed to multiply the left hand side with 1 | σ ` ε p Y n q|“ 1 1 E 4 . 2 n whic h conv erges to one in probabilit y by Corollary 1.5 and Lemma 4.2 . On the even t t| σ ` ε p Y n q| “ 1 u , the righ t eigenspace of Y n asso ciated with λ max p Y n q is one-dimensional. By Lemma 4.1 , w e are therefore reduced to showing that 1 | σ ` ε p Y n q|“ 1 1 E 4 . 2 n ˇ ˇ @ u n , p X n ´ λ max p Y n q I q ´ 1 u n D ˇ ˇ 2 } u n } 2 }p X n ´ λ max p Y n q I q ´ 1 u n } 2 P Ý Ý Ý Ñ n Ñ8 1 ´ 1 | ξ | 2 . Noticing that E 4 . 2 n Ă E 4 . 3 n , we obtain by Lemmas 4.2 and 4.3 that 1 E 4 . 2 n 1 } u n } 2 @ p X n ´ λ max p Y n q I q ´ 1 u n , u n D P Ý Ý Ý Ñ n Ñ8 ´ 1 ξ . By Lemmas 4.2 , 4.4 and 4.6 , it holds that 1 E 4 . 2 n }p X n ´ λ max p Y n q I q ´ 1 u n } } u n } P Ý Ý Ý Ñ n Ñ8 1 a | ξ | 2 ´ 1 , and the result is obtained through a direct calculation. □ 26 EXTREME EIGENV ALUES AND EIGENVECTORS 5. Open problems W e no w presen t several op en problems that emerge naturally from our results. Most of these app ear approach- able using reﬁnements of existing techniques, while one in particular—concerning assumptions on Theorem 1.6 —p oses a more signiﬁcant theoretical challenge and remains largely unresolv ed. Op en Problem 1 (sparser regimes) . The b ounds in ( 2.11 ) and ( 2.4 ) tend to zer o even when K n r emains b ounde d as n Ñ 8 . Our curr ent metho ds alr e ady yield an analo gue of ( 1.3 ) in the c ase K n “ K ą 0 . T o ful ly extend the r esult, one must c ompute the moments of T r p X n q , as in L emma 2.8 . The limiting distribution is not Gaussian—in the dir e cte d Er d˝ os–R´ enyi c ase, for instanc e, the non-Gaussian limit is derive d in [ Cos23 ] . Op en Problem 2 (t yp es of sparsit y) . Extend the analysis to alternative sp arsity r e gimes b eyond that deﬁne d in ( 1.1 ) . F or example, c onsider the Hadamar d pr o duct of an i.i.d. matrix with the adjac ency matrix of a K n -r e gular gr aph, uniformly sample d fr om the sp ac e of such gr aphs. The interplay b etwe en r andomness and structur e d sp arsity pr esents new analytic al chal lenges. Op en Problem 3 (unbounded eigenv alues of E n ) . Pr op osition 2.7 r emains valid if } E n } “ O p n o p 1 q q . Investi- gate whether, after pr op er normalization, the se quenc e q n p z q r emains tight and whether The or em 1.1 c ontinues to hold when } E n } Ñ 8 as n Ñ 8 . Op en Problem 4 (assumptions on Theorem 1.6 ) . Can one r emove the distributional and sp arsity assumptions in The or em 1.6 ? Doing so would r e quir e asymptotic lower b ounds on the le ast singular value s n p X n ´ ξ I q . Our appr o ach dep ends on the universality r esults of [ BvH24 ] , which justify these extr a assumptions. R emoving them app e ars to b e a substantial ly har der pr oblem and is curr ently out of r e ach. Op en Problem 5 (eigenv ectors of ﬁnite-rank p erturbation) . Gener alize The or em 1.6 to the c ase of a defor- mation with an arbitr ary ﬁnite r ank, similarly to what was done in the Hermitian c ase by, e.g. , [ BGN11 ] ). This gener alization is useful for many applic ative c ontexts wher e the matrix E n b e ars an “information ” burie d in the sp arse noise matrix X n . Funding M. Louv aris has received funding from the Europ ean Union’s Horizon 2020 research and innov ation programme under the Marie Sk lodowsk a–Curie grant agreement No. 101034255. A cknowledgement The authors wish to thank Ada Altieri for fruitful discussions. References [ABC ` 24] I. Akjouj, M. Barbier, M. Clenet, W. Hachem, M. Ma ¨ ıda, F. Massol, J. Na jim, and V. C. T ran. Complex systems in ecology: a guided tour with large lotk a–volterra models and random matrices. Pro c e e dings of the R oyal So ciety A , 480(2285):20230284, 2024. [BBAP05] J. Baik, G. Ben Arous, and S. P´ ech ´ e. 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Extreme eigenvalues and eigenvectors for finite rank additive deformations of non-hermitian sparse random matrices

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