Convergence of Half-Space Last Passage Percolation Away from the Boundary to the Directed Landscape

Con v ergence of Half-Space Last P assage P ercolation Aw a y from the Boundary to the Directed Landscap e Xin yi Zhang F ebruary 23, 2026 Abstract In this note, w e pro v e con v ergence of the half-space exp onen tial last passage p ercolation (LPP) mo del, aw ay from the boundary , to the directed landscap e. Our approach couples the half-space and full-space LPP mo dels and constructs t wo barrier ev ents based on the monotonic- it y of last passage paths. Combining this coupling with moderate deviation estimates for b oth mo dels and the kno wn conv ergence of full-space LPP to the directed landscap e, we establish the desired con v ergence. 1 In tro duction The Kardar–P arisi–Zhang (KPZ) universalit y class describ es the large-scale b ehavior of a wide family of random growth and in teracting particle systems in 1 + 1 dimensions. Ov er the past t wo decades, remark able progress has b een made in understanding the full-space KPZ universalit y class. F or geometric and exp onen tial last passage percolation (LPP) models with i.i.d. w eigh ts, Jo- hansson [Joh00] and Baik–Deift–Johansson [BDJ99] pro ved that the properly cen tered and scaled passage time from (0 , 0) to ( n, n ) con verges in distribution to the GUE T racy–Widom la w. Beyond one-p oin t fluctuations, the Airy 2 pro cess was iden tified as the scaling limit of m ulti-p oin t fluctu- ations along the spatial direction by Pr¨ ahofer and Sp ohn [PS02]. Subsequen t works of Corwin, Quastel, and Remenik [CQR13] extended this analysis to contin uum statistics using determinantal structure, while Corwin and Hammond [CH14] established the Brownian Gibbs prop erty of the Airy line ensemble, whose top curv e is the Airy 2 pro cess. More recently , Dauvergne, Ortmann, and Vir´ ag [DOV22] completed this picture by constructing the Airy sheet and the directed landscape, the conjectural universal scaling limits gov erning joint fluctuations in all space-time directions for full-space mo dels within the KPZ class. New phenomena emerge when b oundary conditions are introduced. In the half-space LPP mo del, one considers up-right paths constrained to H = { ( i, j ) ∈ Z 2 : i ≥ j } with differen t diagonal weigh ts representing the b oundary interaction. Baik and Rains [BR00] first disco v ered the crosso ver fluctuation distributions interpolating b etw een the GOE and GUE T racy–Widom distributions, corresp onding to differen t strengths of the b oundary . Later, Baik, Barraquand, Corwin, and Suidan [BBCS18a] identified a tw o-dimensional crosso ver kernel that generalizes the the crossov er distributions found in [FNH99] and [SI04]. In this note w e relate the full-space and half-space LPP mo dels by showing that, when the b oundary is non-attractiv e, the b oundary has no effect in the scaling limit for p oints that lie sufficien tly far inside the bulk. More precisely , we prov e that if the observ ation p oin t is at distance n 2 / 3+ δ from the b oundary for some fixed δ > 0, the half-space last passage time con v erges to the same limit as in the full-space mo del. W e only presen t our pro of for the exp onen tial LPP model, 1 but the same argumen t applies to the geometric LPP as the three k ey mo derate deviation estimates used in the exp onential case all hav e geometric analogues. When the b oundary is non-attractive (in the exp onen tial case, when α ≥ 1 2 ), the full-space and half-space mo dels share the same shap e function, whic h can b e view ed as the la w of large n umber limit of the last passage v alue. Thus, the cost of deviating from the c haracteristic direction b ey ond the transversal fluctuation scale, which is of order n 2 / 3 , to trav el along the b oundary is not comp ensated at leading order. In contrast, when the b oundary b ecomes attractive (in the exp onen tial case, when α < 1 2 ), the shap e function for the half-space model is strictly larger, as the last passage path could b enefit from tra veling along the boundary even at the cost of deviating. In this regime, the p ortion of the path adhering to the b oundary exhibits Gaussian fluctuations. 2 Mo del Definitions and Main Results Let ≺ denote the partial order on Z 2 giv en by ( x 1 , y 1 ) ≺ ( x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≤ y 2 . F or b oth full-space and half-space exp onential LPP with i.i.d. Exp(1) weigh ts and non-attractive b oundary , the length of a t ypical maximal path from ( x 1 , y 1 ) to ( x 2 , y 2 ) is enco ded in the follo wing deterministic shap e function d ( x 1 , y 1 ; x 2 , y 2 ) =  √ x 2 − x 1 + 1 + p y 2 − y 1 + 1  2 , ( x 1 , y 1 ) ≺ ( x 2 , y 2 ) . F or any subset O ⊂ Z 2 , let Π O [( x 1 , y 1 ) → ( x 2 , y 2 )] denote the set of up-righ t lattice paths from ( x 1 , y 1 ) to ( x 2 , y 2 ) contained in O . Definition 2.1 (F ull-space LPP) . Let { w i,j } i,j ∈ Z 2 b e i.i.d. exp onential random v ariables with rate 1. Define the last passage time from ( x 1 , y 1 ) to ( x 2 , y 2 ) as L full ( x 1 , y 1 ; x 2 , y 2 ) = max π ∈ Π[( x 1 ,y 1 ) → ( x 2 ,y 2 )] X ( i,j ) ∈ π \ ( x 2 ,y 2 ) w i,j , if ( x 1 , y 1 ) ≺ ( x 2 , y 2 ) and zero otherwise. Definition 2.2 (Half-space LPP) . Let { w i,j : i, j ∈ Z and i ≥ j } b e indep endently distributed exp onen tial random v ariables with rate 1 if i > j and with rate α if i = j . Define the half space H = { ( i, j ) ∈ Z 2 , i ≥ j } and the half-space last passage time from ( x 1 , y 1 ) to ( x 2 , y 2 ) as L half ( x 1 , y 1 ; x 2 , y 2 ) = max π ⊂ H π ∈ Π[( x 1 ,y 1 ) → ( x 2 ,y 2 )] X ( i,j ) ∈ π \ ( x 2 ,y 2 ) w i,j , if ( x 1 , y 1 ) ≺ ( x 2 , y 2 ) and zero otherwise. Our main result establishes con v ergence to the directed landscap e for half-space exp onential last passage p ercolation n 2 / 3+ δ a wa y from the b oundary . W e do not recall the definition of the directed landscap e here and refer the reader to [DO V22]. Theorem 2.3 (Half-space exp onential LPP conv erges to the directed landscap e) . L et R 4 + = { ( x, s ; r , t ) ∈ R 4 : s < t } . Fix any δ > 0 and define the sc ale d half-sp ac e exp onential last p as- sage p er c olation away fr om the b oundary L half ,δ n as a r andom function on R 4 + : L half ,δ n ( x, s ; y , t ) = 2 − 4 / 3 n − 1 / 3 L half ( ⌊ ns + 2 5 / 3 n 2 / 3 x ⌋ + ⌊ n 2 / 3+ δ ⌋ , ⌊ ns ⌋ ; ⌊ nt + 2 5 / 3 n 2 / 3 y ⌋ + ⌊ n 2 / 3+ δ ⌋ , ⌊ nt ⌋ ) − 2 2 / 3 n 2 / 3 ( t − s ) − 2 4 / 3 n 1 / 3 ( y − x ) . (1) 2 Then L half ,δ n c onver ges to the dir e cte d landsc ap e L in distribution uniformly over c omp act subsets of R 4 + . W e also recall the conv ergence of the full-space mo del [DV22, Theorem 12.1], which pro vides the starting p oint of our analysis. Theorem 2.4. F or the fol lowing sc ale d ful l-sp ac e exp onential last p assage p er c olation, viewe d as a r andom function on R 4 + , L full n ( x, s ; y , t ) = 2 − 4 / 3 n − 1 / 3 L full ( ⌊ ns + 2 5 / 3 n 2 / 3 x ⌋ , ⌊ ns ⌋ ; ⌊ nt + 2 5 / 3 n 2 / 3 y ⌋ , ⌊ nt ⌋ ) − 2 2 / 3 n 2 / 3 ( t − s ) − 2 4 / 3 n 1 / 3 ( y − x ) . (2) we know that L full n c onver ges to the dir e cte d landsc ap e L in distribution uniformly over c omp act subsets on R 4 + . 3 Coupling Bet w een F ull-Space and Half-Space LPP W e couple the full-space mo del and the half-space mo del in the following wa y . Let (Ω , F , P ) b e a probabilit y space supp orting tw o indep endent families of random v ariables { W i,j } ( i,j ) ∈ Z 2 and { U i,i } i ∈ Z , where the { W i,j } are i.i.d. Exp(1) and the { U i,i } are i.i.d. Exp( α ) for some fixed α ≥ 1 2 , indep enden t of { W i,j } . Define the full-space last passage time L full using the w eights { W i,j } ( i,j ) ∈ Z 2 , and the half- space last passage time L half using the weigh ts { W i,j } i>j together with { U i,i } i ∈ Z . Observ e if w e define the scaled full-space exp onen tial last passage p ercolation aw ay from the b oundary as the following: L full ,δ n ( x, s ; y , t ) =2 − 4 / 3 n − 1 / 3 L full ( ⌊ ns + 2 5 / 3 n 2 / 3 x ⌋ + ⌊ n 2 / 3+ δ ⌋ , ⌊ ns ⌋ ; ⌊ nt + 2 5 / 3 n 2 / 3 y ⌋ + ⌊ n 2 / 3+ δ ⌋ , ⌊ nt ⌋ ) − 2 2 / 3 n 2 / 3 ( t − s ) − 2 4 / 3 n 1 / 3 ( y − x ) , (3) then by Theorem 2.4 and the translation inv ariance in distribution of exp onential LPP , w e hav e that L full ,δ con verges to directed landscap e in distribution uniformly o ver compact subsets on R 4 + . By Theorem 2.4, in order to establish the desired con vergence for the half-space mo del, it suffices to prov e the follo wing coupling prop osition: Prop osition 3.1. F or every c omp act D ⊂ R 4 + , lim n →∞ P  L half ,δ n | D  = L full ,δ n | D  = 0 . (4) T o sho w that the probabilit y in equation (4) v anishes, w e will construct t wo barrier even ts whose probability upp er b ound the probability in equation (4) yet still con verges to zero. T o do so, we need the follo wing lemma on the monotonicit y of the righ tmost last passage paths, whose con tinuous version can b e found in [DO V22, Lemma 3.6]. Lemma 3.2. L et O = H or Z 2 . L et ( x 1 , y ) , ( w 1 , z ) , ( x 2 , y ) , ( w 2 , z ) ∈ O such that ( x 1 , y ) ≺ ( w 1 , z ) , ( x 2 , y ) ≺ ( w 2 , z ) , and x 1 ≤ x 2 , w 1 ≤ w 2 . L et P O [( x 1 , y ) → ( w 1 , z )] ⊂ Π O [( x 1 , y ) → ( w 1 , z )] b e the set of p ath π that maximizes the last p assage value in O , i.e., X ( i,j ) ∈ π w i,j = max σ ⊂ O σ ∈ Π[( x 1 ,y ) → ( w 1 ,z )] X ( i,j ) ∈ σ \ ( w 1 ,z ) w i,j . 3 L et π + ( x 1 , y ; w 1 , z ) ∈ P O [( x 1 , y ) → ( w 1 , z )] denote the rightmost maximizing p ath and define the leftmost x -c o or dinate in p ath π on the level j L eftmost j [ π ] = ( i if ( i − 1 , j ) / ∈ π and ( i, j ) ∈ π −∞ if ( i, j ) / ∈ π for al l i ∈ Z (5) Then for any j ∈ Z , L eftmost j  π + ( x 1 , y ; w 1 , z )  ≤ L eftmost j  π + ( x 2 , y ; w 2 , z )  . Pr o of. W rite π 1 := π + ( x 1 , y ; w 1 , z ) and π 2 := π + ( x 2 , y ; w 2 , z ). Assume for con tradiction that there exists j with Leftmost j [ π 1 ] > Leftmost j [ π 2 ]. Planarity implies that there exists t wo p oints ( i 1 , j 1 ) , ( i 2 , j 2 ) ∈ π 1 ∩ π 2 , ( i 1 , j 1 ) ≺ ( i 2 , j 2 ) such that b oth π 1 and π 2 pass through ( i 1 , j 1 ) and ( i 2 , j 2 ) and on the level j ∈ [ j 1 + 1 , j 2 ], π 1 is strictly to the righ t of π 2 . W e truncate the tw o paths into three segmen ts each. Let π 1 = π (1) 1 ∪ π (2) 1 ∪ π (3) 1 where (1) π (1) 1 starts at ( x 1 , y ) and ends at ( i 1 , j 1 ); (2) π (2) 1 starts at ( i 1 + 1 , j 1 ) and ends at ( i 2 , j 2 − 1); (3) π (3) 1 starts at ( i 2 , j 2 ) and ends at ( w 1 , z ). Similarly , write π 2 = π (1) 2 ∪ π (2) 2 ∪ π (3) 2 where (1) π (1) 2 starts at ( x 2 , y ) and ends at ( i 1 , j 1 ); (2) π (2) 2 starts at ( i 1 , j 1 + 1) and ends at ( i 2 − 1 , j 2 ); (3) π (3) 2 starts at ( i 2 , j 2 ) and ends at ( w 2 , z ). Because π 1 is a maximizing path, w e must hav e X ( i,j ) ∈ π (2) 1 w i,j ≥ X ( i,j ) ∈ π (2) 2 w i,j . (6) This implies that π (1) 2 ∪ π (2) 1 ∪ π (3) 2 is also an element in P O [( x 2 , y ) → ( w 2 , z )] and thus con tradicts the fact that π 2 is the rightmost maximizing path. F or any compact set D ∈ R 4 + , we know that D ⊂ [ − M , M ] 4 for some large M ∈ Z . Let us fix ℓ ∈ (0 , 1) and consider n large enough such that n 2 / 3 M < n 2 / 3+ δ (1 − ℓ ). Let ( x 1 , y 1 ) = ( ⌊ ℓn 2 / 3+ δ ⌋ − M n, − M n ) and ( x 2 , y 2 ) = ( ⌊ ℓn 2 / 3+ δ ⌋ + M n, M n ). Let Diagonal = { ( i, i ) : i ∈ Z } and A ( H ) = { ω ∈ Ω : for every π ∈ P H [( x 1 , y 1 ) → ( x 2 , y 2 )] , π ∩ Diagonal  = ∅} A ( Z 2 ) = { ω ∈ Ω : for every π ∈ P Z 2 [( x 1 , y 1 ) → ( x 2 , y 2 )] , π ∩ Diagonal  = ∅} . (7) On the complemen t of A ( H ) ∪ A ( Z 2 ), there exists a maximizing path π ⋆ ∈ P Z 2 [( x 1 , y 1 ) → ( x 2 , y 2 )] and a maximizing path π ⋆ ∈ P H [( x 1 , y 1 ) → ( x 2 , y 2 )] which a void the diagonal. By Lemma 3.2, we see that all the righ tmost maximizing paths b et ween endp oin ts ( x, s ) and ( y , t ) suc h that ( x, s ; y , t ) ∈ D a void the diagonal. By the construction of our coupling, the full-space and half-space mo dels use iden tical weigh ts at all off-diagonal sites. Therefore, for all ( x, s ; y , t ) ∈ D , L half ( x, s ; y , t ) = L full ( x, s ; y , t ) on the complement of A ( H ) ∪ A ( Z 2 ). This implies P  L half ,δ n | D  = L full ,δ n | D  ≤ P  A ( H ) ∪ A ( Z 2 )  ≤ P ( A ( H )) + P ( A ( Z 2 )) . 4 4 Mo derate Deviation Estimates W e will state and derive a few mo derate deviation results in this section in order to upp er b ound the probabilities P ( A ( H )) and P ( A ( Z 2 )). The following theorem comes from [LR10, Theorem 2] and is rephrased in terms of exp onential last passage time in [BHS22, Theorem 2.2] Prop osition 4.1 (F ull-space one-p oint mo derate deviation) . F or any K > 1 , ther e exists two strictly p ositive c onstants C ( K ) , c ( K ) such that for al l m, n ≥ 1 with K − 1 < m n < K and al l r > 0 we have: P  L full (0 , 0; m, n ) − ( √ m + √ n ) 2 ≥ rn 1 / 3  ≤ C e − c min { r 3 / 2 ,rn 1 / 3 } (8) P  L full (0 , 0; m, n ) − ( √ m + √ n ) 2 ≤ − rn 1 / 3  ≤ C e − cr 3 . (9) The next result giv es an upper b ound on the low er tail of constrained last passage v alue. Let us first define a cylinder C [( w , z ) , ξ , γ ] of width 2 √ 2 γ n 2 / 3 from ( w , z ) to ( w + ξ n, z + ξ n ) in Z 2 : C γ  ( w , z ) , ξ  := n ( x, y ) ∈ Z 2 : 2 z ≤ x + y ≤ 2 z + 2 ξ n, z − w − 2 γ n 2 / 3 ≤ − x + y ≤ z − w + 2 γ n 2 / 3 o . (10) Moreo ver, let L C γ ( w , z ; w + ξ n, z + ξ n ) denote the maximum w eight o ver all path from ( w , z ) to ( w + ξ n, z + ξ n ) constrained inside C γ [( w , z ) , ξ ]. No w we are ready to introduce the follo wing prop osition from [BGHH22, Prop osition 3.7]: Prop osition 4.2 (F ull-space constrained mo derate deviation) . Fix L 1 , L 2 > 0 . L et L 1 ≤ γ ≤ L 2 . Then ther e exists p ositive c onstants θ 0 , n 0 , which dep end on L 1 , L 2 , and an absolute c onstant c > 0 such that for n > n 0 and θ > θ 0 , P  L C γ (1 , 1; ξ n, ξ n ) − 4 ξ n ≤ − θ n 1 / 3  ≤ e − cγ θ . (11) Lastly , w e need to deriv e an upp er b ound on the upp er tail of half-space exp onential LPP . W e will use the k ey observ ation in [BBCS18b] that the CDF of the half-space exp onential LPP can b e written as a F redholm Pfaffian. Let us first introduce the definition of a F redholm Pfaffian. Definition 4.3. Let K ( x, y ) b e a 2 × 2 matrix-v alued k ernel on a measure space ( X , µ ), written as K ( x, y ) = K 11 ( x, y ) K 12 ( x, y ) K 21 ( x, y ) K 22 ( x, y ) ! , x, y ∈ X . Let J ( x, y ) b e defined by J ( x, y ) = 1 x = y 0 1 − 1 0 ! . W e say that K is skew-symmetric if, for every collection of p oints x 1 , . . . , x 2 k ∈ X , the blo ck matrix ( K ( x i , x j )) 2 k i,j =1 is sk ew-symmetric. The F r e dholm Pfaffian of K is defined, whenever the series con verges, by Pf ( J + K ) L 2 ( X ,µ ) = 1 + ∞ X k =1 1 k ! Z X k Pf  K ( x i , x j )  k i,j =1 dµ ( x 1 ) · · · dµ ( x k ) . 5 F or a finite 2 k × 2 k sk ew-symmetric matrix A = ( a ij ), its Pfaffian is giv en by Pf ( A ) = 1 2 k k ! X σ ∈ S 2 k sgn( σ ) a σ (1) σ (2) a σ (3) σ (4) · · · a σ (2 k − 1) σ (2 k ) . Additionally , w e need the following lemma [BBCS18b, Lemma 2.5] which is prov ed by Hadamard’s inequalit y and the fact that Pf ( A ) = p det( A ) for any sk ew-symmetric matrix A . Lemma 4.4. L et K ( x, y ) b e a 2 × 2 skew-symmetric matrix-value d kernel. Supp ose ther e exist c onstants C > 0 and r e al numb ers a > b ≥ 0 such that | K 11 ( x, y ) | ≤ C e − a ( x + y ) , | K 12 ( x, y ) | ≤ C e − ax + by , | K 22 ( x, y ) | ≤ C e b ( x + y ) . Then, for every inte ger k > 0 ,   Pf  K ( x i , x j )  k i,j =1   ≤ (2 k ) k/ 2 C k k Y i =1 e − ( a − b ) x i . Finally , we are ready to state the k ey proposition [BBCS18b, Prop osition 1.6] and [BBCS18b, Lemma 6.4]: Prop osition 4.5. F or al l α ≥ 1 / 2 , r ∈ R and every p ositive inte ger m , P  L half (1 , 1; m, m ) − 4 m < 2 4 / 3 m 1 / 3 r  = Pf [ J − K exp ,m ] L 2 ( r, ∞ ) . (12) F or α = 1 / 2 , the hyp otheses of L emma 4.4 ar e satisfie d for m lar ge enough. The exact form ulas K exp ,m is slightly different for the case that α > 1 2 and the case that α = 1 2 . Since we will not use the explicit form of K exp ,m , we omit it for simplicit y . W e no w use Lemma 4.4 to con trol the conv ergence of the F redholm Pfaffian series and to derive a quan titative upper b ound for the tail of L half (1 , 1; m, m ). Prop osition 4.6 (Half-space upp er tail estimate) . F or al l α ≥ 1 / 2 , ther e exist c onstants c, C > 0 and m 0 ∈ N such that, for al l m ≥ m 0 and al l r > 0 , P  L half (1 , 1; m, m ) − 4 m ≥ m 1 / 3 r  ≤ C e − cr . Pr o of. By Prop osition 4.5, we know that P  L half (1 , 1; m, m ) − 4 m ≥ 2 4 / 3 m 1 / 3 r  = 1 − Pf [ J − K exp ,m ] L 2 ( r, ∞ ) . where this Pfaffian can b e expanded as a conv ergen t series: 1 − Pf [ J − K exp ,m ] L 2 ( r, ∞ ) = − ∞ X k =1 ( − 1) k k ! Z ( r, ∞ ) k Pf  K exp ,m ( x i , x j )  k i,j =1 dx 1 · · · dx k . Lemma 4.4 guarantees that for α = 1 / 2, eac h term of this expansion satisfies the b ound    Pf  K exp ,m ( x i , x j )  k i,j =1    ≤ (2 k ) k/ 2 C k k Y i =1 e − ( a − b ) x i , 6 where the constants a > b ≥ 0 are indep endent of m . Integrating this b ound term by term yields    Z ( r, ∞ ) k Pf  K exp ,m ( x i , x j )  k i,j =1 dx 1 · · · dx k    ≤ (2 k ) k/ 2 C k ( a − b ) − k e − k ( a − b ) r . Since the resulting series is absolutely summable, we obtain 1 − Pf [ J − K exp ,m ] L 2 ( r, ∞ ) ≤ ∞ X k =1 1 k ! (2 k ) k/ 2 C k e − k ( a − b ) r ≤ C ′ e − ( a − b ) r , for some C ′ > 0. Lastly , since exp onential random v ariables are sto chastically ordered in their rate parameter, L half (1 , 1; m, m ) with α > 1 / 2 is sto c hastically dominated b y L half (1 , 1; m, m ) with α = 1 / 2. Thus, the same upp er tail applies. 5 Con v ergence Awa y from the Boundary Recall the endp oints ( x 1 , y 1 ) =  ⌊ ℓn 2 / 3+ δ ⌋ − M n, − M n  , ( x 2 , y 2 ) =  ⌊ ℓn 2 / 3+ δ ⌋ + M n, M n  , with fixed M ∈ Z > 0 , fixed ℓ ∈ (0 , 1), and n large enough that n 2 / 3 M < n 2 / 3+ δ (1 − ℓ ). W e now prov e the following inequality for the shap e function, sho wing that deviations b eyond the typical transversal scale n 2 / 3 pro duce a gap b etw een the free energy at leading order. Lemma 5.1. L et I = [ ⌊ ℓn 2 / 3+ δ ⌋ − M n, M n ] . Ther e exist p ositive c onstants n 0 , c 0 , c 1 such that the fol lowing holds for al l n ≥ n 0 . (i) F or any i, j ∈ I satisfying i ≤ j ≤ i + n 1 / 3 , we have d ( x 1 , y 1 ; j, j ) + d ( i, i ; x 2 , y 2 ) ≤ 8 M n − c 0 n 1 / 3+2 δ . (13) (ii) F or any i, j, s, t ∈ I satisfying i ≤ j ≤ s ≤ t , t ≤ s + n 1 / 3 , and j ≤ i + n 1 / 3 , we have d ( x 1 , y 1 ; j, j ) + d ( i, i ; t, t ) + d ( s, s ; x 2 , y 2 ) ≤ 8 M n − c 1 n 1 / 3+2 δ . (14) Pr o of. One can easily chec k that for A > max { 0 , B } , p A 2 − B 2 ≤ A − B 2 2 A . Then, d ( x 1 , y 1 ; j, j ) = ( p j − x 1 + 1 + p j − y 1 + 1) 2 = 2 j − x 1 − y 1 + 2 + p (2 j − x 1 − y 1 + 2) 2 − ( y 1 − x 1 ) 2 ≤ 2(2 j − x 1 − y 1 + 2) − ⌊ ℓn 2 / 3+ δ ⌋ 2 2(2 j − x 1 − y 1 + 2) . Similarly , d ( i, i ; x 2 , y 2 ) ≤ 2( x 2 + y 2 − 2 i + 2) − ⌊ ℓn 2 / 3+ δ ⌋ 2 2( x 2 + y 2 − 2 i + 2) . Therefore, d ( x 1 , y 1 ; j, j ) + d ( i, i ; x 2 , y 2 ) ≤ 8 M n + 4 j − 4 i − ⌊ ℓn 2 / 3+ δ ⌋ 2 4 M n + 2 ≤ 8 M n + 4 n 1 / 3 − c 0 n 1 / 3+2 δ . W e absorb the term 4 n 1 / 3 in to the constant c 0 b y choosing n 0 large enough. The pro of for (14) follo ws analogously . 7 Pr o of of Pr op osition 3.1. If ω ∈ A ( Z 2 ), then there exists a maximal path π ( ω ) ∈ P Z 2 [( x 1 , y 1 ) → ( x 2 , y 2 )] such that ( i, i ) ∈ π ( ω ) for some i ∈ [ ⌊ ℓn 2 / 3+ δ ⌋ − M n, M n ] . By the definition of last passage path, we know that L full ( x 1 , y 1 ; x 2 , y 2 ) ≤ L full ( x 1 , y 1 ; i, i ) + L full ( i, i ; x 2 , y 2 ) . (15) Let us consider n large enough so that n 1 / 3 > m 0 for the constant m 0 in Prop osition 4.6. Let w 0 < w 1 < · · · < w k ( n ) b e a sequence of in tegers suc h that w 0 = ⌊ ℓn 2 / 3+ δ ⌋ − M n, w 1 = ⌊ ℓn 2 / 3+ δ ⌋ − M n + ⌊ n 1 / 3 ⌋ , w k ( n ) − 1 = M n − ⌊ n 1 / 3 ⌋ and w k ( n ) = M n . Moreov er, m 0 ≤ w j +1 − w j ≤ n 1 / 3 for all 0 ≤ j ≤ k ( n ) − 1. If w j ≤ i < w j +1 , then L full ( x 1 , y 1 ; i, i ) + L full ( i, i ; x 2 , y 2 ) ≤ L full ( x 1 , y 1 ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) . (16) No w w e are ready to b ound P ( A ( Z 2 )). Recall that L C γ ( w , z ; w + ξ n, z + ξ n ) denote the maximum w eight ov er all path from ( w , z ) to ( w + ξ n, z + ξ n ) constrained inside C γ [( w , z ) , ξ ]. Cho ose γ > θ 0 for the constant θ 0 in Prop osition 4.2 and n large enough such that 8 γ n 2 / 3 < ℓn 2 / 3+ δ . P ( A ( Z 2 )) ≤ P  L C γ ( x 1 , y 1 ; x 2 , y 2 ) < max 0 ≤ j ≤ k ( n ) − 1 n L full ( x 1 , y 1 ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) o  ≤ P  L C γ ( x 1 , y 1 ; x 2 , y 2 ) − 8 M n < − 1 2 c 0 n 1 / 3+2 δ  + P  max 0 ≤ j ≤ k ( n ) − 1 n L full ( x 1 , y 1 ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) o − 8 M n > − 1 2 c 0 n 1 / 3+2 δ  ≤ P  L C γ ( x 1 , y 1 ; x 2 , y 2 ) − 8 M n < − 1 2 c 0 n 1 / 3+2 δ  + 2 M n max 0 ≤ j ≤ k ( n ) − 1 P  L full ( x 1 , y 1 ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) − 8 M n > − 1 2 c 0 n 1 / 3+2 δ  By Prop osition 4.2, we kno w that P  L C γ ( x 1 , y 1 ; x 2 , y 2 ) − 8 M n < − 1 2 c 0 n 1 / 3+2 δ  ≤ e − cn 2 δ for some constant c and all n large enough. On the other hand, b y Lemma 5.1, we ha ve P  L full ( x 1 , y 1 ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) − 8 M n > − 1 2 c 0 n 1 / 3+2 δ  ≤ P  L full ( x 1 , y 1 ; w j +1 , w j +1 ) − d ( x 1 , y 1 ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) − d ( w j , w j ; x 2 , y 2 ) > 1 2 c 0 n 1 / 3+2 δ  ≤ P  L full ( x 1 , y 1 ; w j +1 , w j +1 ) − d ( x 1 , y 1 ; w j +1 , w j +1 ) > 1 4 c 0 n 1 / 3+2 δ  + P  L full ( w j , w j ; x 2 , y 2 ) − d ( w j , w j ; x 2 , y 2 ) > 1 4 c 0 n 1 / 3+2 δ  . (17) Let K > 1 to b e the constan t in Prop osition 4.1 that will b e chosen later. If K − 1 ≤ y 2 − w j x 2 − w j , w j +1 − x 1 w j +1 − y 1 ≤ K , then w e can directly apply Prop osition 4.1 and get an upper b ound of C e − c min { n 3 δ ,n 1 / 3+2 δ } . If 8 not, we apply the following cheap b ound P  L full (1 , 1; m, m ′ ) − d (1 , 1; m, m ′ ) > 0  ≤  m ′ + m − 2 m − 1  P m + m ′ X i =1 W i − m − m ′ > 2 √ mm ′ ! ≤ exp  log  m ′ + m − 2 m − 1  − √ mm ′  where W i are i.i.d. Exp(1) random v ariables. W e use the fact that E [ e W 1 / 2 ] = 2 and exponential Mark ov inequality in the last inequality . In our case, w e ha ve m, m ′ ≥ n 1 / 3 and 0 < m ′ m < K − 1 . By Stirling’s approximation, log  m + m ′ − 2 m − 1  ≤ m  1 + m ′ m  log  1 + m ′ m  − m ′ m log  m ′ m  + O (log m ) . Th us, we can choose K − 1 suc h that log  m ′ + m − 2 m − 1  − √ mm ′ ≤ − √ mm ′ / 2 ≤ − n 1 / 3 / 2. Thus, (17) is upp er b ounded by C e − c min { n 2 δ ,n 1 / 3 } . No w we will pro ve the analogous statemen t for the half-space cas e. Let L half | full ( x 1 , y 1 ; i, i ) = max π ⊂ H ; π ∩ Diagonal=( i,i ) π ∈ Π[( x 1 ,y 1 ) → ( i,i )] X ( u,v ) ∈ π \ ( i,i ) w u,v L half | full ( i, i ; x 2 , y 2 ) = max π ⊂ H ; π ∩ Diagonal=( i,i ) π ∈ Π[( i,i ) → ( x 2 ,y 2 )] X ( u,v ) ∈ π \ ( i,i ) , ( x 2 ,y 2 ) w u,v Let L R i,j = L half | full ( x 1 , y 1 ; w i +1 , w i +1 ) + L half ( w i , w i ; w j +1 , w j +1 ) + L half | full ( w j , w j ; x 2 , y 2 ) , L i,j = L full ( x 1 , y 1 ; w i +1 , w i +1 ) + L half ( w i , w i ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) . Since L i,j ≥ L R i,j for all 0 ≤ i ≤ j ≤ k ( n ) − 1, w e can now b ound P ( A ( H )) by the follo wing P  L C γ ( x 1 , y 1 ; x 2 , y 2 ) < max 0 ≤ i ≤ j ≤ k ( n ) − 1 L R i,j  ≤ P  L C γ ( x 1 , y 1 ; x 2 , y 2 ) < max 0 ≤ i ≤ j ≤ k ( n ) − 1 L i,j  ≤ P  L C γ ( x 1 , y 1 ; x 2 , y 2 ) − 8 M n < − 1 2 c 1 n 1 / 3+2 δ  + 4 M 2 n 2 max 0 ≤ i ≤ j ≤ k ( n ) − 1 P  L i,j − 8 M n > − 1 2 c 1 n 1 / 3+2 δ  (18) W e again apply Prop osition 4.2 to b ound the first term. F or the second term, we apply Lemma 5.1. P  L full ( x 1 , y 1 ; w i +1 , w i +1 ) + L half ( w i , w i ; w j +1 , w j +1 ) + L full ( w j , w j ; x 2 , y 2 ) − 8 M n > − 1 2 c 1 n 1 / 3+2 δ  = P  L full ( x 1 , y 1 ; w i +1 , w i +1 ) − d ( x 1 , y 1 ; w i +1 , w i +1 ) > 1 6 c 1 n 1 / 3+2 δ  + P  L half ( w i , w i ; w j +1 , w j +1 ) − d ( w i , w i ; w j +1 , w j +1 ) > 1 6 c 1 n 1 / 3+2 δ  + P  L full ( w j , w j ; x 2 , y 2 ) − d ( w j , w j ; x 2 , y 2 ) > 1 6 c 1 n 1 / 3+2 δ  (19) 9 Then, by Prop osition 4.1, Prop osition 4.6, and the c heap b ound ab o v e, the ab ov e probability is b ounded by C e − c min { n 2 δ ,n 1 / 3 } . 6 Ac kno wledgmen ts The author sincerely thanks their advisor, Iv an Corwin, for constan t guidance and supp ort. The author is esp ecially grateful to Sa y an Das for suggesting this problem, and to Milind Hegde for detailed feedbac k on an early draft of this work. The author also thanks Jiyue Zeng and Alan Zhao for v aluable conv ersations. 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