Stabilization time of finite configurations with a second class particle in discrete TASEP

Stabilization time of finite configurations with a second class particle in discrete T ASEP Bori Anna Mészáros ∗ Bálin t V ető † Marc h 31, 2026 Abstract W e consider finite configurations of particles and holes sampled according to Bernoulli pro duct measure and with a second class particle added to a random p osition. The stabilization time is the num ber of steps needed to reach an ordered state under discrete time T ASEP dynamics with parallel up date. W e describ e the additional time of stabilization caused b y the presence of the second class particle. 1 In tro duction The totally asymmetric simple exclusion pro cess (T ASEP) is an interacting particle sys- tem with configurations on the in teger lattice Z where eac h lattice p oint is either occupied or v acant. It was first introduced in contin uous time in [ Spi70 ], the discrete time version w as studied e.g. in [ Sc h93 ]. Our in terest is in the dynamics with parallel up date where particles jump to the righ t by one indep enden tly with a fixed probabilit y q if the target p osition is empt y . See also [ MS11 ] for other p ossible discrete time T ASEP dynamics. The particle dynamics can b e describ ed in terms of the heigh t function determined mo dulo a constan t global shift by particles and holes corresp onding to segmen ts with slop e − 1 and +1 resp ectively . P article jumps mean that pairs of segments with slop e − 1 on the left and with slop e +1 on the right are swapped with probabilit y q turning the lo cal maxim um of the heigh t in to a lo cal minimum. The T ASEP evolution on finite segments of the lattice corresp onds to the swap dy- namics of finite sequences of tw o t yp es. The stabilization time of a string is the n um b er of steps from the initial configuration to the final state under the swap dynamics with q = 1 and with parallel up date. The finiteness of the stabilization time for any initial string is an exercise in [ UM996 ] in terpreted in terms of a line of soldiers facing in random directions. F or pro duct Bernoulli initial condition with density p ∈ [0 , 1] , the asymptotic distribution of the stabilization time w as describ ed in [ FNN15 ] as the length of the string tends to infinity , see Theorem 1.1 b elo w. In the presen t pap er, w e study the stabilization time of random sequences with three t yp es. In addition to particles and holes, the third t yp e is the second class particle which ∗ Departmen t of Sto chastics, Institute of Mathematics, Budap est Universit y of T ec hnology and Eco- nomics, Műegy etem rkp. 3., H-1111 Budap est, Hungary . E-mail: meszaros.bori@edu.bme.hu † Departmen t of Sto chastics, Institute of Mathematics, Budap est Universit y of T ec hnology and Eco- nomics, Műegyetem rkp. 3., H-1111 Budapest, Hungary and HUN–REN Alfréd Rényi Institute of Math- ematics, Reáltanoda u. 13–15., H-1053 Budapest, Hungary . E-mail: vetob@math.bme.hu 1 w as defined for T ASEP in [ FK95 ] and serv ed as the main to ol to b ound the curren t fluctuations and diffusivity in ASEP in [ BS10 ]. Parallel up date is not a priori well-defined with second class particles b ecause of p ossible conflicts. W e resolv e the conflicts by the con v ention that the dynamics for three t yp es reduces to that for tw o t yp es under the pro jection whic h replaces second class particles b y holes. W e compare the full stabilization time for three types to that for the pro jected dynamics in the presence of one second class particle. Started from pro duct Bernoulli initial strings together with the lo cation of the second class particle, we describ e the asymptotic distribution of the additional time in the limit for long strings. 1.1 Stabilization with t w o t yp es W e introduce the sorting mo del for binary strings that corresp onds to discrete time T ASEP with parallel up date, and w e recall the result on the stabilization time of random strings from [ FNN15 ]. Let Ω (2) n = { 0 , 2 } n b e the set of strings of length n with tw o t yp es. W e define the ev olution step Z (2) : Ω (2) n → Ω (2) n in a wa y that it sw aps neighouring elemen ts whic h are in increasing order, that is, every o ccurrence of 02 is replaced by 20 within the string. Replacing each 0 with a particle and each 2 with a hole, Z (2) describ es a discrete time step of T ASEP with parallel up date. Since no tw o o ccurrences of 02 can o verlap, the definition do es not result in con- flicts b etw een o verlapping pairs. The successive application of Z (2) ev en tually sorts the elemen ts in non-increasing order, and the pro cess stabilizes in at most n − 1 steps. Let T (2) n ( ω ) = min { k ≥ 0 : ( Z (2) ) k ( ω ) = ( Z (2) ) k +1 ( ω ) } (1.1) denote the stabilization time for the binary sequence ω ∈ Ω (2) n where ( Z (2) ) k is the k fold application of Z (2) . The natural measure on Ω (2) n is the Bernoulli pro duct measure where eac h element of the string is 2 with probabilit y p and 0 with probabilit y 1 − p indep endently . The main result in [ FNN15 ] iden tifies the asymptotic fluctuations of the stabilization time T (2) n as n → ∞ dep ending on p . Theorem 1.1. F or p ∈ (0 , 1) \ { 1 / 2 } fixe d, we have that T (2) n − max( p, 1 − p ) n √ n d = ⇒ N (0 , p (1 − p )) (1.2) in distribution as n → ∞ wher e the limit is Gaussian with me an 0 and varianc e p (1 − p ) . F or p = 1 / 2 , T (2) n − n/ 2 √ n d = ⇒ χ 3 2 (1.3) in distribution as n → ∞ wher e χ 3 is the chi distribution with p ar ameter 3 . F or p = 1 / 2 + λ/ (2 √ n ) with λ ∈ R fixe d, we have T (2) n − n/ 2 √ n d = ⇒ M λ 1 − 1 2 B λ 1 (1.4) in distribution as n → ∞ wher e B λ t is a Br ownian motion with drift λ and M λ t = max s ∈ [0 ,t ] B λ s is its running maximum pr o c ess. 2 1.2 Stabilization mo del with three t yp es The purp ose of this paper is to inv estigate the case of three types. Let Ω n = { 0 , 1 , 2 } n denote the space of strings of length n of three t yp es. The ev olution step Z : Ω n → Ω n w ould ideally sw ap all p ossible increasing substrings 01 , 02 , 12 in to their decreasing coun terparts. Ho w ever, these substrings can o verlap and form the substring 012 . T o resolv e the conflict, one has to give priority to one of the tw o o v erlapping substrings. In this pap er, we use the conv ention that the substring 12 has priority ov er 01 and changes first into 21 , that is, Z ( . . . 012 . . . ) = . . . 021 . . . . (1.5) Ha ving this settled, the evolution step Z is w ell-defined. In terms of a particle system, 0 corresp onds to a particle, 2 is a v acan t site, and 1 is a second class particle. Under the T ASEP dynamics, the particles jump to the righ t if their righ t neighbour is v acant. Second class particles also jump to the right to v acant lo cations but they can also jump to the left if there is a particle on their left. The t w o p ossible jumps of second class particles can lead to a conflict in di screte time with parallel up date if three consecutiv e sites contain a particle, a second class particle and a hole. W e resolve this conflict b y giving priority to the right jumps of the second class particle whic h is the priorit y rule giv en in ( 1.5 ). F ormally , if ω i = 0 , ω i +1 = 1 , ω i +2 = 2 for some i = 1 , . . . , n − 2 then ( Z ω ) i = 0 , ( Z ω ) i +1 = 2 , ( Z ω ) i +2 = 1 . F urthermore, if ω i − 1 ≥ ω i < ω i +1 ≥ ω i +2 for some i = 1 , . . . , n − 1 (with the inequalit y alwa ys considered satisfied for ω 0 and ω n +1 ) then ( Z ω ) i = ω i +1 , ( Z ω ) i +1 = ω i . The tw o natural pro jections Π 1 , Π 2 : Ω n → Ω (2) n are given by (Π i ω ) j = π i ( ω j ) (1.6) for i = 1 , 2 and j = 1 , . . . , n and ω ∈ Ω n where π 1 (0) = π 1 (1) = 0 , π 1 (2) = 2 and π 2 (0) = 0 , π 2 (1) = π 2 (2) = 2 . In other words, the tw o t yp es under Π 1 are { 0 , 1 } and { 2 } , that is, w e consider 0 and 1 iden tical. Under Π 2 , the t yp es are { 0 } and { 1 , 2 } . There is no ev olution rule for three types under whic h b oth natural pro jections evolv e according to the dynamics for t w o t yp es. With our con v en tion, the evolution for three types pro jected b y Π 1 follo ws the dynamics of tw o types, that is, Π 1 Z = Z (2) Π 1 but it do es not hold for Π 2 instead of Π 1 . Starting from any string ω ∈ Ω n , the rep eated applications of the evolution step Z ev en tually stabilize, and the pro cedure results in a string where all elements are sorted in non-increasing order. The stabilization time T n ( ω ) is the num b er of steps needed to ac hiev e the non-increasing order starting from ω . F ormally , w e let T n ( ω ) = min { k ≥ 0 : Z k ( ω ) = Z k +1 ( ω ) } . (1.7) In the example 0122102 Z − → 0212120 Z − → 2021210 Z − → 2202110 Z − → 2220110 Z − → 2221010 Z − → 222110 (1.8) the stabilization time is 6 , that is, T 7 (0122102) = 6 . In this paper, we consider a sp ecial initial distribution of the string ω ∈ Ω n with three t yp es where there is a single element of type 1 i n ω . The initial string ω is constructed in t w o steps as follows. First, we sample the locations of 2 s according to the Bernoulli 3 pro duct measure of parameter p , that is, we let ω ′ ∈ Ω n b e a string where each elemen t is independently equal to 2 with probability p and is 0 with probabilit y 1 − p . Then conditionally given the set A ( ω ′ ) of lo cations of 0 s in ω ′ , w e sample U with uniform distribution on the set A ( ω ′ ) . Then w e c hange the corresp onding element ω ′ U in to 1 to obtain ω . If A ( ω ′ ) = ∅ , then we let ω 1 = 1 . W e refer to the dis tribution of ω as the Bernoulli initial condition of parameter p with a second class particle in uniform p osition. 1.3 Main results The stabilization time result in Theorem 1.1 describ es the asymptotic b ehaviour of T (2) n (Π 1 ( ω )) for an y string ω ∈ Ω n with three types. After T (2) n (Π 1 ( ω )) steps in a string ω ∈ Ω n , all the 2 s are sorted to the b eginning of the string but there might still b e some 0 s b efore the p osition of 1 in the string, whic h need to b e sw app ed with 1 . Hence T (2) n (Π 1 ( ω )) is not equal to the total stabilization time T n ( ω ) . W e denote by E n the n um b er of excess steps, formally , w e let E n ( ω ) = T n ( ω ) − T (2) n (Π 1 ( ω )) . (1.9) Our first main result is the following c haracterization of the excess for fixed p . Theorem 1.2. L et p ∈ (0 , 1) b e fixe d and let ω ∈ Ω n b e sample d ac c or ding to the Bernoul li initial c ondition of p ar ameter p with a se c ond class p article in uniform p osition. Then the exc ess E n c onver ges in distribution E n d = ⇒    1 if p > 1 / 2 , X if p = 1 / 2 , 0 if p < 1 / 2 (1.10) as n → ∞ wher e the distribution of X is given by P ( X = 0) = P ( X = 1) = 1 / 2 . The second main result describ es the limiting distribution of the excess E n in the case when p is critically scaled with n around 1 / 2 . Theorem 1.3. L et p = 1 2 − λ 2 √ n for some λ ∈ R b e fixe d and let ω ∈ Ω n b e sample d ac c or ding to the Bernoul li initial c ondition of p ar ameter p with a se c ond class p article in uniform p osition. Then we have that E n d = ⇒ X ( λ ) (1.11) as n → ∞ wher e the distribution of X ( λ ) is given by P ( X ( λ ) = 0) = 1 − P ( X ( λ ) = 1) = E (argmax s ∈ [0 , 1] B ( λ ) s ) (1.12) with a Br ownian motion B ( λ ) s with drift λ . The pap er is organized as follows. W e characterize the p ossible initial p ositions of the 1 in a string which result in a p ositiv e excess in Section 2 by pro viding a structural understanding of the ev olution of heigh ts. The main results are pro ved in Section 3 using hitting time b ounds for random walks. W e prov e these b ounds in Section 4 . 4 k S k S M S M − 1 S M − 2 K = M 2 M 1 M 0 = M Figure 1: An example illustrating the p oin ts M 0 = M , M 1 , and K = M 2 . A c kno wledgmen ts. The w ork of B. A. Mészáros was supp orted by the NKFI (National Researc h, Dev elopmen t and Inno v ation Office) grant FK142124. The work of B. V ető was supp orted by the NKFI grants ADV ANCED 150474 and KKP144059 “F ractal geometry and applications”. 2 Switc hing p osition for the excess The pro ofs of Theorems 1.2 and 1.3 are based on understanding how the initial p osition of the element 1 determines the excess. In particular, in this section, we giv e a structural understanding of the ev olution of lo cal maxima which leads to a c haracterization of the initial p ositions of 1 with excess. F or a string ω with a unique 1 , the maximal prefix of 2 s and the maximal suffix of 0 s are L ( ω ) = max { k : (Π 1 ω ) 1 = (Π 1 ω ) 2 = · · · = (Π 1 ω ) k = 2 } , R ( ω ) = max { k : (Π 1 ω ) n = (Π 1 ω ) n − 1 = · · · = (Π 1 ω ) n − k +1 = 0 } . (2.1) The height function of a string with tw o types consists of ± 1 steps. F or a string ω ∈ Ω n with three types, we consider the height function corresp onding to the pro jection Π 1 ( ω ) giv en b y S k = k X i =1 (1 − (Π 1 ω ) i ) (2.2) for k = 0 , 1 , . . . , n . W e introduce K ( ω ) = max { k ∈ { L + 2 , . . . , n − R − 1 } : S l < S k − 1 , l = L, . . . , k − 2 } (2.3) as the rightmost p osition of the string ω with the prop erty that the height remains strictly b elo w S K − 1 b et w een L and K − 2 and we let K = L + 2 if the set on the righ t-hand side of ( 2.3 ) is empty . See Figure 1 for an illustration. By definition ( 2.3 ), w e hav e π 1 ( ω K − 2 ) = π 1 ( ω K − 1 ) = π 1 ( ω K ) = 0 and π 1 ( ω K +1 ) = 2 unless K = L + 2 . By the next result, K is the switching p osition for the p ositivit y of the excess in the sense that the excess E n is zero if and only if the initial p osition of the 1 is b efore K . In addition, the excess can b e more than one only when the initial p osition of the 1 is to the right of n − R , that is, there is no 2 after it. 5 Prop osition 2.1. F or any string ω with a single 1 in it, we have the e quality of events { U ( ω ) < K ( ω ) } = { E n ( ω ) = 0 } . (2.4) F urthermor e, { E n ( ω ) > 1 } ⊆ { U ( ω ) > n − R ( ω ) } . (2.5) Let M k denote the leftmost lo cation with height that is k less than the maxim um of S k on { L + 1 , . . . , n − R − 1 } , that is, M k ( ω ) = min { j ∈ { L + 1 , . . . , n − R − 1 } : S j = max { S i : i ∈ { L + 1 , . . . , n − R − 1 } − k } (2.6) for k = 0 , 1 , 2 , . . . whic h are well-defined as long as the set of indices j ab ov e is not empt y . W e mention that M 0 ( ω ) is the p osition of the leftmost maximum of S k for k ∈ { L + 1 , . . . , n − R − 1 } and w e also denote it b y M = M 0 . See Figure 1 . It follows from the definition that π 1 ( ω M k ) = π 1 ( ω M k − 1 ) = 0 unless M k = L + 1 and that S j < S M k for j = L, . . . , M k − 1 . F or a given ω , we hav e K ( ω ) = M m ( ω ) for some in teger m by definition ( 2.3 ) unless the set of indices k is empt y on the righ t-hand side of ( 2.3 ). It is imp ossible to hav e M k − 1 ( ω ) = M k ( ω ) + 1 for some k = 1 , . . . , m b ecause it w ould mean that K ( ω ) ≥ M k − 1 ( ω ) . Hence the p oints M 0 ( ω ) , M 1 ( ω ) , . . . , M m ( ω ) decomp ose the interv al [ K , M ] in to blo cks of integer lengths that are all at least 3 . The idea of the pro of is that for the given ω , we keep track of the p oint M k for k = 0 , . . . , m under the evolution step Z applied to ω rep eatedly . W e introduce ω ( j ) = Z j ω , L ( j ) = L ( Z j ω ) , R ( j ) = R ( Z j ω ) , M ( j ) k = M k ( Z j ω ) (2.7) and we let U ( j ) denote the p osition of 1 in ω ( j ) . Evolution o ccurs in three phases. The first phase is the bulk b ahaviour of M k . F or any k = 0 , . . . , m , we ha ve M ( j +1) k = M ( j ) k − 1 (2.8) if M ( j ) k ≥ L ( j ) + 2 . Since M ( j ) k − 1  = M ( j ) k + 1 , M ( j ) k is a local maximum of the height, that is, π 1 ( ω M ( j ) k ) = 0 and π 1 ( ω M ( j ) k +1 ) = 2 . On the other hand, all local maxima in [ L ( j ) , M ( j ) k − 1] ha v e heights lo w er than S M ( j ) k b y ( 2.6 ). Applying the evolution step ω 7→ Z ω decreases all the lo cal maxima of the height, resulting in that M ( j ) k − 1 b ecomes a new leftmost lo cal maxim um with v alue S M ( j +1) 0 − k , that is, ( 2.8 ) holds. The second phase is the e dge b ahaviour whic h can happ en in later steps of the evolu- tion. If M ( j ) k = L ( j ) + 1 and M ( j ) k − 1 > L ( j ) + 2 for some j = 1 , 2 , . . . , then we hav e M ( j +1) k = M ( j ) k + 1 . (2.9) Indeed, if M ( j ) k is a leftmost lo cal maximum with v alue S M ( j ) 0 − k then π 1 (( ω ( j ) ) M ( j ) k ) = 0 and π 1 (( ω ( j ) ) M ( j ) k +1 ) = 2 hold. The pair 02 is transformed into 20 , so the height at M ( j ) k ± 1 b ecomes S M ( j +1) 0 − k = S M ( j ) 0 − k − 1 but M ( j ) k − 1 = L ( j +1) whic h pro ves ( 2.9 ). The final phase is the annihilation . If M ( j ) k = L ( j ) + 1 and M ( j ) k − 1 = L ( j ) + 2 then we necessarily hav e π 1 (( ω ( j ) ) L ( j ) +1 ) = π 1 (( ω ( j ) ) L ( j ) +2 ) = 0 and π 1 (( ω ( j ) ) L ( j ) +3 ) = 2 since M k − 2 6 can only b e in the bulk b ehaviour phase. This pro jected pattern 002 after the left blo ck of 2 s is up dated in the next step to 020 , resulting in having M ( j +1) k − 1 = M ( j ) k − 1 − 1 (2.10) and in the disapp earance of M k . Pr o of of Pr op osition 2.1 . First, w e giv e the pro of of ( 2.4 ) for the case when K ( ω ) > L ( ω ) + 2 . W e assume that U ( ω ) < K ( ω ) . As seen ab o v e, K ( ω ) = M m ( ω ) and we also ha v e that M m +1 ( ω ) = K ( ω ) − 1 . F urthermore, applying the evolution step Z rep eatedly , M m +1 follo ws the bulk b ehaviour describ ed ab ov e, that is, go es to the left b y one in ev ery step as long as M ( j ) m +1 = M ( j ) m − 1 ≥ L ( j ) + 2 . If this condition holds then we ha v e π 1 (( ω ( j ) ) M ( j ) m − 2 ) = π 1 (( ω ( j ) ) M ( j ) m − 1 ) = π 1 (( ω ( j ) ) M ( j ) m ) = 0 and π 1 (( ω ( j ) ) M ( j ) m +1 ) = 2 . It corresp onds to a pro jected string 0002 that mo ves to the left by one in every step during the bulk b eha viour of M m +1 . If U ( ω ) ≤ M m ( ω ) − 1 = K ( ω ) − 1 holds for the initial p osition of the 1 then its later p osition cannot exceed that of the second 0 of the mo ving string 0002 during the bulk phase, that is, U ( j ) ≤ M ( j ) m +1 as long as M ( j ) m +1 ≥ L ( j ) + 2 . The last step of the bulk b eha viour of M m +1 happ ens if M ( j ) m +1 = L ( j ) + 2 when the 0002 string meets the initial string of 2 s of length L ( j ) . The num b er of initial 2 s remains the same in this step, the string 0002 transforms in to 0020 and M m +1 annihilates immediately after the end of its bulk b ehaviour. If U ( ω ) ≤ K ( ω ) − 1 holds initially then the only p ossible p osition of the 1 after j + 1 steps is at L ( j +1) + 1 , that is, the string 0020 must b e the pro jection of 1020 . Since the 1 is to the left of all 0 s in ω ( j +1) , after the sorting of the pro jected string Π 1 ( ω ( j +1) ) , the sorting of ω ( j +1) will also ha v e b een completed, whic h means that E n ( ω ) = 0 . Next, we assume that U ( ω ) ≥ K ( ω ) . W e sho w that if U ( ω ) > M k ( ω ) for some k = 1 , . . . , m then the 1 remains on the right of M k un til the annihilation of M k . During the bulk b eha viour of M k , the 1 cannot mov e to the left of M k trivially . Assume that M k arriv es at the edge after j steps. Then we hav e M ( j ) k = L ( j ) + 1 and π 1 (( ω ( j ) ) M ( j ) k ) = 0 but ( ω ( j ) ) M ( j ) k = 1 is imp ossible if U ( ω ) > M k ( ω ) . In further steps, the string 02 at p ositions M k , M k + 1 transforms into 20 , hence M k mo v es to the right by one according to ( 2.9 ) and ( ω ( j +1) ) M ( j +1) k = 0 also holds. This means that the 1 remains on the righ t of M k during the edge phase. The annihilation of M k happ ens when M ( j ) k = M ( j ) k − 1 − 1 = L ( j ) + 1 which corresp onds to π 1 (( ω ( j ) ) L ( j ) +1 ) = π 1 (( ω ( j ) ) L ( j ) +2 ) = 0 and π 1 (( ω ( j ) ) L ( j ) +3 ) = 2 , and the pro jected string 002 is turned into 020 . If the 1 was betw een M k and M k − 1 , that is, U ( j ) = M ( j ) k − 1 then the string 012 at p ositions L ( j ) + 1 to L ( j ) + 3 is transformed into 021 . This means that the 1 is transferred to the right of M k − 1 and U ( j +1) = M ( j +1) k − 1 + 2 = L ( j +1) + 3 . If U ( ω ) = K ( ω ) then M m +1 ( ω ) = K ( ω ) − 1 which means U ( ω ) = M m ( ω ) for the p osition of 1 which remains true during the bulk phase of M m +1 . The end of the bulk phase means that U ( j ) = M ( j ) m = M ( j ) m +1 + 1 = L ( j ) + 3 for some j and the corresp onding string is 0012 after the initial string of 2 s of length L ( j ) . In the next step, M m +1 annihilates, the string transforms into 0021 and we hav e M ( j +1) m = M ( j ) m − 1 = L ( j ) + 2 and U ( j +1) = L ( j ) + 4 , that is, the 1 is transferred to the right of M m . By induction, if U ( ω ) ≥ K ( ω ) then the 1 is transferred to the righ t of M ( j ) 0 for some j . F rom then on, the 1 remains on the righ t of M 0 during its bulk and edge phases as seen ab o ve. The last step of the edge phase of M 0 o ccurs when M ( j ) 0 = L ( j ) + 1 = n − R ( j ) − 1 for some j after whic h Π 1 ( ω ( j +1) ) b ecomes sorted. Since the 1 w as to the right of M ( j ) 0 , it 7 cannot b e at p osition L ( j +1) + 1 in ω ( j +1) whic h means that E n ( ω ) ≥ 1 . This completes the pro of of ( 2.4 ) for the case K ( ω ) > L ( ω ) + 2 . If K ( ω ) = L ( ω ) + 2 , then U ( ω ) < K ( ω ) necessarily implies that 1 is to the left of all 0 s already in the initial string ω and that w e hav e E n ( ω ) = 0 . In the case U ( ω ) ≥ K ( ω ) , w e ha ve M k ( ω ) = L ( ω ) + 1 for some integer k and the 1 is to the righ t of M k whic h results in E n ( ω ) ≥ 1 by the ab ov e argument. Finally , w e prov e ( 2.5 ) b y showing that U ( ω ) ≤ n − R ( ω ) implies E n ( ω ) ≤ 1 . If U ( ω ) ≤ n − R ( ω ) holds then the last 2 at p osition n − R ( ω ) initially is swapped with the 1 in some step j , that is, the string ω ( j ) ends with 21 and a string of 0 s of length R ( j ) . In any further ev olution step after this, the last 2 can only mo v e to the left by swapping with a 0 . If this happ ens then the 1 is sw app ed with the same 0 in the next step. The distance of the last 2 and the 1 cannot exceed 2 , hence after the last swap of the last 2 , at most one more sorting step is needed, which yields E n ( ω ) ≤ 1 . 3 Limiting excess distribution In this section, we pro ve our main results stated in Theorems 1.2 and 1.3 . The pro ofs are based on computin g the exp ectation of the switc hing p osition K . The k ey observ ation is that the difference of the switc hing p osition K and the p osition of the leftmost maximum M can b e controlled in the sense that their exp ected distance can b e upp er b ounded as in Prop osition 3.1 b elo w. The pro of of Prop osition 3.1 is p ostp oned to Section 4 . Prop osition 3.1. 1. F or p = 1 / 2 − λ/ (2 √ n ) , ther e is a c onstant c that dep ends only on λ such that for al l n almost sur ely E  M − K   M  ≤ c √ n. (3.1) 2. F or p < 1 / 2 fixe d, ther e is a deterministic c onstant c such that E  M − K   M  ≤ c almost sur ely for al l n . Prop osition 3.2. F or p > 1 / 2 , we have that lim n →∞ E ( M ) n = 0 . (3.2) Pr o of of Pr op osition 3.2 . The regime p > 1 / 2 is where the drift of the random w alk S k is 1 − 2 p < 0 negative. W e first sho w that if S 0 = 0 then for an y ε > 0 , we ha v e that lim n →∞ P 0  max k ≥ ε √ n S k > 0  = 0 . (3.3) where the subindex 0 means conditioning on { S 0 = 0 } . Since m n = E 0 ( S ε √ n ) = (1 − 2 p ) ε √ n < 0 , the cen tral limit theorem implies that the fluctuations of S ε √ n around its mean m n are of order n 1 / 4 . In particular, lim n →∞ P 0  S ε √ n > m n 2  = 0 . (3.4) By conditioning on the v alue of S ε √ n , we can write P 0  max k ≥ ε √ n S k > 0  = P 0  max k ≥ ε √ n S k > 0    S ε √ n > m n 2  P 0  S ε √ n > m n 2  + P 0  max k ≥ ε √ n S k > 0    S ε √ n < m n 2  P 0  S ε √ n < m n 2  (3.5) 8 The first pro duct of probabilities on the righ t-hand side of ( 3.5 ) go es to 0 by ( 3.4 ). In the second pro duct, the conditional probability can b e upp er b ounded by a | m n | / 2 where a = P 0 ( ∃ k : S k = 1) < 1 since the drift of S k is negative. This prov es ( 3.3 ). The limit in ( 3.3 ) means that argmax k ∈{ 1 ,...,n } S k √ n P → 0 (3.6) as n → ∞ in probability conditionally giv en that S 0 = 0 . The argmax ma y not b e unique for the random w alk S m but ( 3.6 ) holds for any of them. The v alue of M is equal to the p osition of the leftmost maximum of the random walk started at L + 1 . The v ariable L has a geometric distribution with parameter 1 − p and in p articular it do es not dep end on n . Hence M √ n P → 0 (3.7) also holds as n → ∞ in probabilit y . Since M ≤ n almost surely , ( 3.7 ) implies ( 3.2 ). Pr o of of The or em 1.2 . Let first p > 1 / 2 b e fixed. By Prop osition 2.1 , the asymptotic excess distribution can b e obtained from the probability P ( U < K ) ≤ P ( U < M ) (3.8) where the upp er b ound follo ws b ecause K ≤ M holds almost surely . W e compute P ( U < M ) as follows. The num b er of 0 s in the first m bits of the string is equal to ( m + S m ) / 2 . Since U is a uniformly chosen p osition among the 0 s, for any fixed m , we hav e that P  U < m   Π 1 ω  = m + S m n + S n . (3.9) The fluctuations of the random w alk S m with drift 1 − 2 p around its mean are of order √ n . More precisely , for any ε > 0 there is a C large enough so that P  max m ∈{ 1 ,...,n } | S m − m (1 − 2 p ) | ≤ C √ n  > 1 − ε (3.10) for all n . By conditioning, we can write P ( U < M ) = E  P  U < M   Π 1 ω  = E  M + S M n + S n  . (3.11) By ( 3.10 ), the denominator in the exp ectation on the righ t-hand side of ( 3.11 ) is asymp- totically equal to 2(1 − p ) n with a correction of order √ n with probabilit y at least 1 − ε . On the same ev en t with probability at least 1 − ε , the n umerator is 2(1 − p ) M with a correction of order √ n . Then Prop osition 3.2 implies that the righ t-hand side of ( 3.11 ) go es to 0 as n → ∞ . This together with Prop osition 2.1 and the b ound ( 3.8 ) show that lim n →∞ P ( E n = 0) = 0 . Since R has geometric distribution with parameter p which does not dep end on n , we also hav e lim n →∞ P ( E n > 1) = 0 whic h concludes the pro of for p > 1 / 2 . F or p < 1 / 2 fixed, we argue by time rev ersal and w e first sho w that lim n →∞ E ( M ) n = 1 . (3.12) 9 It do es not directly follow from Prop osition 3.2 since M is the p osition of the leftmost maxim um but it is clear from the pro of that the statement holds for the p osition of an y maxim um. Then the same argumen t applies as for p > 1 / 2 and in particular ( 3.10 ) and ( 3.11 ) hold. These together show that lim n →∞ P ( U < M ) = 1 . By Prop osition 2.1 , P ( E n = 0) = P ( U < K ) = P ( U < M ) − P ( U ∈ [ K , M )) (3.13) where we used in the last equality that K ≤ M alw a ys holds. By the second part of Prop osition 3.1 , the exp ected difference E ( M − K ) remains b ounded in n hence P ( U ∈ [ K, M )) go es to 0 as n → ∞ which prov es that lim n →∞ P ( E n = 0) = 1 . The statement for p = 1 / 2 is a sp ecial case of Theorem 1.3 whic h is sho wn b elow. Pr o of of The or em 1.3 . Let p = 1 / 2 − λ/ (2 √ n ) with λ ∈ R fixed. By Prop osition 2.1 , the asymptotic excess distribution can b e obtained from the probability in ( 3.13 ). By Prop osition 3.1 , the exp ected difference of K and M is of order √ n , hence the probabilit y P ( U ∈ [ K, M )) on the righ t-hand side of ( 3.13 ) go es to 0 as 1 / √ n in the n → ∞ limit. W e compute P ( U < M ) as in ( 3.9 ). By the fluctuation result ( 3.10 ), the denominator in the exp ectation on the right-hand side of ( 3.11 ) is asymptotically equal to n with fluctuations of order √ n with high probability . The numerator in the same formula is M with a correction of order √ n with high probability . Hence by b oundedness, w e ha ve P ( U < M ) ∼ E ( M ) n (3.14) as n → ∞ . Since Brownian motion with a drift has a unique argmax almost surely , Donsk er’s in v ariance principle and the contin uous mapping theorem together imply that the argmax M of the random walk S m con v erges to that of the Bro wnian motion with drift λ . 4 Hitting time b ounds This section is dev oted to the pro of of P rop osition 3.1 on the exp ected distance of K and M . Their exp ected distance can b e b ounded b y reading the string ω from M bac kw ards conditionally giv en M . and by applying hitting time estimates for random walks stated in Lemma 4.1 b elow. Since M is the p osition of the leftmost maxim um, the distribution of the remaining steps is that of a random walk conditioned not to visit its starting p osition again. The tra jectory of this walk can then b e decomp osed in to excursions defined b y the returns to different levels b elow the maximum. T o state our random w alk estimates, we define Y n to b e a simple random w alk starting at Y 0 = 0 and with i.i.d. steps P ( Y n +1 − Y n = − 1) = p and P ( Y n +1 − Y n = 1) = q = 1 − p where p ∈ (0 , 1) . W e let V k = min { n = 0 , 1 , . . . : Y n = k } (4.1) the time of the first visit at level k for any k ∈ Z . Lemma 4.1. 1. In the symmetric c ase p = 1 / 2 , we have that P 2  V 1 < m   V 0 > m  → 1 2 (4.2) E 2  V 1 1 { V 1 ≤ m }   V 0 > m  ∼ r π 2 √ m (4.3) as m → ∞ . 10 2. If p = 1 2  1 − λ √ n  then ther e ar e c onstants c 1 > 0 and c 2 finite which dep end on λ such that c 1 < P 2  V 1 < m   V 0 > m  < 1 − c 1 (4.4) E 2  V 1 1 { V 1 ≤ m }   V 0 > m  ≤ c 2 √ m (4.5) holds for al l m ≤ n . Pr o of of L emma 4.1 . F or general p ∈ (0 , 1) , the distribution of the hitting time is giv en b y P 1 ( V 0 = 2 k + 1) = 1 2 k + 1  2 k + 1 k  p k +1 q k (4.6) for all k = 0 , 1 , . . . which can b e computed by the reflection principle and it can also b e found in [ F el68 ]. The generating function of the probability distribution in ( 4.6 ) is g ( s ) = 1 − p 1 − 4 pq s 2 2 q s . (4.7) The probability that the hitting do es not happ en in finite time can b e given as P 1 ( V 0 = ∞ ) = 1 − g (1) whic h is equal to 0 for p ≥ 1 / 2 and w e hav e P 1 ( V 0 = ∞ ) = ( q − p ) /q for p < 1 / 2 . W e sp ecify p = 1 2  1 − λ √ n  and we use the notation P ( λ ) and E ( λ ) to denote the dep endence on λ in this pro of where P (0) and E (0) corresp onds to the p = 1 / 2 case. By Stirling’s approximation, the asymptotics of the probabilit y in ( 4.6 ) is giv en b y P ( λ ) 1 ( V 0 = 2 k + 1) ∼ 1 2 √ π k 3 / 2 e − 2 λ 2 k/n (4.8) as k → ∞ . The tail b eha viour of the hitting time can b e computed exactly for p = 1 / 2 b y P (0) 1 ( V 0 > m ) ∼ r 2 π 1 √ m (4.9) as m → ∞ whic h follows b y summation in ( 4.8 ). F or general λ ∈ R fixed, we hav e that e c 1 √ m ≤ P ( λ ) 1 ( V 0 > m ) ≤ e c 2 √ m (4.10) in the join t limit when m, n → ∞ pro vided that m ≤ n . The constan ts e c 1 , e c 2 are p ositiv e and finite and they only dep end on λ . The upp er b ound in ( 4.10 ) follows by the comparison with the symmetric λ = 0 case. The asymptotic v alue of P ( λ ) 1 ( V 0 = 2 k + 1) is upp er b ounded b y P (0) 1 ( V 0 = 2 k + 1) using ( 4.8 ) for k large. On the other hand, P ( λ ) 1 ( V 0 = ∞ ) = λ/ √ n for all λ > 0 but this is still O (1 / √ m ) . F or the low er b ound, we sum the right-hand side of ( 4.8 ) b et ween m and 2 m only which results in m terms all b eing at least a constant times 1 /m 3 / 2 . Next w e expand the left-hand side of ( 4.2 ) and ( 4.12 ) in general. By the definition of the conditional probability and by using the law of total probability , one can write P ( λ ) 2  V 1 < m   V 0 > m  = P m/ 2 − 1 k =0 P ( λ ) 2 ( V 1 = 2 k + 1) P ( λ ) 1 ( V 0 > m − 2 k − 1) P m/ 2 − 1 k =0 P ( λ ) 2 ( V 1 = 2 k + 1) P ( λ ) 1 ( V 0 > m − 2 k − 1) + P ( λ ) 2 ( V 1 > m ) . (4.11) 11 On the other hand, E ( λ ) 2  V 1 1 { V 1 ≤ m }   V 0 > m  = m/ 2 − 1 X k =0 (2 k + 1) P ( λ ) 2  V 1 = 2 k + 1   V 0 > m  = m/ 2 − 1 X k =0 (2 k + 1) P ( λ ) 2 ( V 1 = 2 k + 1) P ( λ ) 1 ( V 0 > m − 2 k − 1) P ( λ ) 2 ( V 0 > m ) . (4.12) Next we turn to the pro of of ( 4.2 ). F or λ = 0 , the asymptotics of the n umerator of the righ t-hand side of ( 4.11 ) can be obtained by splitting the sum at m 1 − ε for some small ε > 0 . F or the first part of the sum, w e hav e m 1 − ε X k =0 P (0) 2 ( V 1 = 2 k + 1) P (0) 1 ( V 0 > m − 2 k − 1) ∼ r 2 π 1 √ m (4.13) b ecause in this regime the probabilities P (0) 1 ( V 0 > m − 2 k − 1) ∼ q 2 π 1 √ m for all 0 ≤ k ≤ m 1 − ε with an additiv e error of order 1 /m 1 / 2+ ε uniformly . The sum of the probabilities P (0) 2 ( V 1 = 2 k + 1) add up to P (0) 2 ( V 1 ≤ 2 m 1 − ε + 1) = 1 − O (1 /m 1 / 2 − ε/ 2 ) . The second part of the sum satisfies m/ 2 − 1 X k = m 1 − ε P (0) 2 ( V 1 = 2 k + 1) P (0) 1 ( V 0 > m − 2 k − 1) = O  1 m 1 − 3 ε/ 2  (4.14) since we can b ound P (0) 2 ( V 1 = 2 k + 1) = O (1 /m 3(1 − ε ) / 2 ) . W e also hav e m/ 2 − 1 X k = m 1 − ε P (0) 1 ( V 0 > m − 2 k − 1) ∼ r 2 π m/ 2 − m 1 − ε X j =1 1 √ 2 j = O ( √ m ) (4.15) whic h pro ves ( 4.14 ). Putting together ( 4.13 ) and ( 4.14 ) and using it in ( 4.11 ) along with ( 4.9 ) yields ( 4.2 ). Using the asymptotics ( 4.8 ) with λ = 0 and ( 4.9 ) in ( 4.12 ), w e get E ( λ ) 2  V 1 1 { V 1 ≤ m }   V 0 > m  ∼ m/ 2 − 1 X k =0 (2 k + 1) 1 2 √ π k 3 / 2 · q 2 π 1 √ m − 2 k − 1 q 2 π 1 √ m ∼ 1 √ π √ m 1 m m/ 2 − 1 X k =0 1 q k m 1 q 1 − 2 k +1 m ∼ r π 2 √ m (4.16) since 1 m m/ 2 − 1 X k =0 1 q k m 1 q 1 − 2 k +1 m → Z 1 / 2 0 1 √ x 1 √ 1 − 2 x d x = π √ 2 (4.17) as m → ∞ whic h prov es ( 4.3 ). 12 The pro ofs of ( 4.4 ) and ( 4.5 ) follow by comparison with their λ = 0 counterparts ( 4.2 ) and ( 4.5 ). The input for proving ( 4.2 ) and ( 4.3 ) are the asymptotics of the hitting time distribution giv en in ( 4.8 ) with λ = 0 and in ( 4.9 ). These asymptotics are then used in ( 4.11 ) and ( 4.12 ). Let λ ∈ R b e fixed. The expansions ( 4.11 ) and ( 4.12 ) are still v alid. The asymptotic expansion ( 4.8 ) is no w used with the giv en v alue of λ . Since w e assume that m ≤ n , the asymptotics only change by constan t factors. More precisely , instead of the asymptotics in ( 4.8 ), we can giv e upp er and low er b ounds with t wo differen t p ositive and finite constant prefactors. Similarly , instead of the asymptotics ( 4.9 ), the b ounds giv en in ( 4.10 ) are a v ailable for the tail of the hitting time. Using these asymptotic results, the b ounds in ( 4.4 ) and ( 4.5 ) follo w in the same w ay as the asymptotics in the λ = 0 case. Pr o of of Pr op osition 3.1 . W e start with a detailed proof of the first part ab out the p = 1 / 2 − λ/ (2 √ n ) case. In the end, we show ho w the statemen t for fixed p < 1 / 2 follows. Conditionally given M and L , we introduce the rev ersed and reflected random walk e S i seen from the p oin t ( M , S M ) given by e S i = S M − S M − i (4.18) for i = 0 , 1 , . . . , M − L . By the definition ( 4.18 ) and b y the leftmost maximum prop erty of M , it follo ws that the la w of e S i is that of a simple random w alk with drift 2 p − 1 starting at e S 0 = 0 conditioned not to return to 0 until the ( M − L ) th step. In particular w e ha ve S 1 = 1 and S 2 = 2 . Next w e in tro duce a sequence of indicators I k and the corresp onding random times τ k . W e let τ 0 = 1 and we let I 0 = 1 ( e S τ 0 + i > e S τ 0 − 1 for i = 1 , . . . , M − L − τ 0 ) which is almost surely equal to 1 since e S i cannot return to 0 . Next we define τ 1 = τ 0 + 1 = 2 . In general, given τ k , we let I k = 1  e S τ k + i > e S τ k − 1 for i = 1 , . . . , M − L − τ k  (4.19) for k = 1 , 2 , . . . , that is, I k is the indicator that the random walk ( e S τ k + i , i = 1 , 2 , . . . , M − L − τ k ) stays strictly ab ov e the level e S τ k − 1 . If I k = 1 then we let τ k +1 = τ k + 1 and in this case we hav e e S τ k +1 = e S τ k + 1 . If I k = 0 then we define τ k +1 = min n i ∈ { 1 , . . . , M − L − τ k } : e S τ k + i = e S τ k − 1 o + 1 (4.20) where the minimum ab ov e is the time of the first return to the level e S τ k − 1 which is finite b y the fact that I k = 0 . The addition of 1 on the right-hand side of ( 4.20 ) ensures that e S τ k +1 = e S τ k . W e let the k th excursion last b et w een times τ k and τ k +1 with length W k = τ k +1 − τ k . The pair ( τ k , I k ) forms a Marko v c hain and its transition probabilities are describ ed b elo w. F or deterministic t k , t k +1 ∈ Z + and i k , i k +1 ∈ { 0 , 1 } , one can write the conditional probabilit y P (( t k , i k ) , ( t k +1 , i k +1 )) = P  ( τ k +1 , I k +1 ) = ( t k +1 , i k +1 )   ( τ k , I k ) = ( t k , i k )  (4.21) dep ending on i k as follo ws. If i k = 1 then the transition probability P (( t k , 1) , ( t k +1 , i k +1 )) is nonzero only if t k +1 = t k + 1 . In that case we hav e P (( t k , 1) , ( t k + 1 , 1)) = 1 − P (( t k , 1) , ( t k + 1 , 0)) = P 2  V 1 > M − L − t k +1   V 0 > M − L − t k +1  (4.22) 13 since b etw een times τ k +1 and M − L , the w alk e S j is conditioned not to go strictly b elow e S τ k +1 − 1 . If i k = 0 then the distribution of τ k +1 is given by P (( t k , 0) , ( t k +1 , { 0 , 1 } )) = P 2  V 1 = t k +1 − t k   V 1 ≤ M − L − t k < V 0  (4.23) since the random walk e S j conditioned not to go strictly b elo w e S τ k − 1 b et w een times τ k and M − L do es visit e S τ k − 1 and the time of the first visit is τ k +1 with distribution given b y the righ t-hand side of ( 4.23 ). F urther, given that τ k +1 = t k +1 , we hav e P  I k +1 = 1   τ k +1 = t k +1  = 1 − P  I k +1 = 0   τ k +1 = t k +1  = P 2  V 1 > M − L − t k +1   V 0 > M − L − t k +1  (4.24) b ecause the w alk e S j b et ween τ k +1 and M − L is conditioned not to go strictly b elo w e S τ k +1 − 1 . Then the transition probabilities P (( t k , 0) , ( t k +1 , i k +1 )) are given by the pro ducts of the probabilities in ( 4.23 ) and ( 4.24 ). With the definition W k = τ k +1 − τ k b eing the length of the k th excursion, w e let N = min { k = 1 , 2 , . . . : I k = I k − 1 = 1 } (4.25) to b e the first time when there are t wo consecutiv e trivial excursions whic h is a stopping time. Then w e can write M − K + 1 = N X k =1 W k (4.26) b y the definition of K . Since N is a stopping time for the time inhomogeneous Marko v c hain ( τ k , I k ) and for its natural filtration F k , the computation similar to the pro of of W ald’s identit y yields E N X k =1 W k ! = ∞ X k =1 E  W k 1 { N ≥ k }  = ∞ X k =1 E  E  W k 1 { N ≥ k }   F k − 1  = ∞ X k =1 E  E  W k   F k − 1  1 { N ≥ k }  . (4.27) The conditional exp ectation E ( W k |F k − 1 ) is b ounded by a constan t multiple of √ n uni- formly in k by Lemma 4.1 . The same lemma also implies that N has an exp onential tail and in particular E ( N ) is b ounded b y a constan t. Hence the exp ectation of ( 4.26 ) is upp er b ounded by a constant times √ n which completes the proof of ( 3.1 ) and that of the first part. If p < 1 / 2 is fixed then the same decomp osition into excursions of length W k remain v alid. The difference is that the n umber and the exp ected lengths of excursions is now related to those of a random w alk with a fixed p ositiv e drift. Instead of Lemma 4.1 w e ha v e that P ( V 0 = ∞ ) > 0 and V 1 has an exp onen tial tail. Hence w e hav e that lim m →∞ P 2  V 1 < m   V 0 > m  = P 2  V 1 < ∞   V 0 = ∞  = p q q − p q 1 −  p q  2 = p. (4.28) 14 On the other hand, the conditional exp ectation E 2  V 1 1 { V 1 ≤ m }   V 0 > m  is b ounded by a constant uniformly in m . These b ounds are enough to conclude the second part of the prop osition. References [BS10] M. Balázs and T. Seppäläinen. 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[UM996] Univ ersit y of Michigan Undergraduate Mathematics Comp etition 13. https : / / lsa . umich . edu / content / dam / math - assets / math - document / umumc / UMUMC13.pdf , 1996. 15