arXiv:math/0605212v3 [math.PR] 18 Oct 2007 iekre,srn prxmto,ReanHletprobl Riemann–Hilbert approximation, mials. strong kernel, sine atpsaeproainmdl ihgoercadexponen and geometric with en models ran matrix percolation of random passage subsequence of last increasing terms [ longest in spec tations the expressible of include number not p Examples a are statistical of and models laws theory these probability limit combinatorics, describe in also els theory matrix dom distribu universal probability (different) gap a to the converges while limit” distribution, scaling converg Tracy–Widom eigenvalue largest measu the the of to large, class becomes general matrix very the me a of probability For a conjugation. with unitary equipped under matrices Hermitian of set the 1807–1834 5, No. DOI: 35, Vol. Probability2007, of Annals The nrno arxter r needn ftepeiestruc precise the of independent [ ensemble are the of theory randomness matrix random in
c aiainadtpgahcdetail. typographic 1807–1834 and 5, the pagination by No. published 35, Vol. article 2007, original Statistics the Mathematical of of reprint Institute electronic an is This nvriyo ihgn n ro n nvriyo Californ of University and Arbor Ann Michigan, of University nttt fMteaia Statistics Mathematical of Institute thsbe icvrdta h iiigdsrbtosaris distributions limiting the that discovered been has It e od n phrases. and words Key 2 1 .Introduction. 1. eevdArl2006. April Received M 00sbetclassification. subject 2000 AMS upre npr yNFGat M-2050adDMS-05-534 and Fel DMS-02-02530 Sloan Grants the NSF and by DMS-04-57335 part Grant in NSF Supported by part in Supported 10.1214/009117906000001105 ADMMTI ETA II HOESFOR THEOREMS LIMIT CENTRAL MATRIX RANDOM 5 admHriinmtie stedmnino h arcsg matrices the of eigenva dimension the infinity. as the limitin behave to as several mid-time infinity, matrices the Hermitian to at random grow walks steps the time of of number tributions of func the number where generating regime the moment limiting and certain finite a a in that have prove We increments the that tion , 13 ecnie oitretn admwlsstsyn h co the satisfying walks random nonintersecting consider We , yJnoBaik Jinho By OITRETN ADMWALKS RANDOM NONINTERSECTING 31 , 43 ti nw htvroslmtn oa ttsisarising statistics local limiting various that known is It ,rno ze n eao iigmdl [ models tiling Hexagon and Aztec random ], oitretn admwls rc–io distribution, Tracy–Widom walks, random Nonintersecting 2007 , 11 hsrpitdffr rmteoiia in original the from differs reprint This . 60F05. , 1 17 n ofcM Suidan M. Toufic and in , Cruz 18 h naso Probability of Annals The 1 , 21 , 32 , 44 , 48 m tete–ietpolyno- Stieltjes–Wigert em, .Freape consider example, For ]. yis apparently, hysics; tion. si distribution in es 2 lowship. , 03. sr invariant asure e,a h size the as res, usof lues dis- g walks o permu- dom tion. nte“bulk the in rows ndi- ilrandom tial n nran- in ing ueo the of ture a Santa ia, fi mod- ific sembles. 9 , 33 ], 2 J. BAIK AND T. M. SUIDAN variables [30], polynuclear growth models [34, 45] and vicious walker mod- els [3, 27]. For these models, the distribution function of interest was com- puted explicitly in terms of certain determinantal formulae and the asymp- totic analysis of these determinants yielded the desired limit law. Neverthe- less, it is believed that such limit laws should hold for a class of models much wider than the explicitly computable (“integrable”) models. One such universality result for models “outside random matrices” was obtained in [10, 12, 50] for thin last passage percolation models with general random variables. This paper studies nonintersecting random walks and proves random ma- trix central limit theorems in a certain limiting regime. The motivation for this study comes from two sources. The first is the fact that the eigenvalue density function of the Gaussian unitary ensemble can be described in terms of a nonintersecting Brownian bridge process [22, 33]. Namely, consider n (1) (n) standard Brownian bridge processes (Bt ,...,Bt ) conditioned not to in- (1) (n) tersect during the time interval (0, 2) (i.e., Bt > >Bt for 0
1.1. Motivating examples. We begin by introducing two distribution func- tions. Define the kernels
Ai(a)Ai′(b) Ai′(a)Ai(b) sin(π(a b)) (1) A(a, b)= − , S(a, b)= − . a b π(a b) − − Set A S (2) FTW(ξ) = det(1 (ξ, )), FSine(η) = det(1 [ η,η]). − | ∞ − | − The Tracy–Widom distribution, FTW, is the limiting distribution of the largest eigenvalue and FSine is the limiting distribution for the gap proba- bility of the eigenvalues “in bulk” in Hermitian random matrix theory.
(1) (n) 1.1.1. Nonintersecting Brownian bridge process. Let Bt = (Bt ,...,Bt ) be an n-dimensional standard Brownian motion. We compute the density function of B conditioned that B(1) >B(2) > >B(n) for 0 2 B = B = (0,..., 0). Let p (x,y)= 1 e (x y) /(2t). The argument of Kar- 0 2 t √2πt − − lin and McGregor [36] implies that the density function of n one-dimensional nonintersecting Brownian motions at time t which start from (x1,...,xn), where x > >x , is given by 1 · · · n (3) f (b , . . ., b ) = det(p (x , b ))n , b > > b . t 1 n t i j i,j=1 1 · · · n Hence, for b > > b , the density function of B equals 1 · · · n 1 n n det(p1(xi, bj))i,j=1 det(p1(bi,yj))i,j=1 f(b1, . . . , bn) = lim · x,y 0 det(p (x ,y ))n → 2 i j i,j=1 (4) n(n 1)/2 n 2 − 2 b2 = b b e− j , n/2 n 1 i j π j=1− j! 1 i ℓn(π) 2√n (6) lim P − 1.1.3. Symmetric simple random walks and random rhombus tilings of a hexagon. Consider n symmetric simple (Bernoulli) random walks S(m)= (S(1)(m),...,S(n)(m)), conditioned not to intersect and such that S(0) = (2(n 1), 2(n 2),..., 0) = S(2k). Any realization of such walks is in one- to-one− correspondence− with a rhombus tiling of a hexagon with side lengths k,k,n,k,k,n. Again, using the argument of Karlin and McGregor, the distri- bution of S(k) can be expressed in terms of a determinant. This determinant was significantly simplified and was shown to be related to the so-called Hahn orthogonal polynomials by Johansson [33]. A further asymptotic analysis of the Hahn polynomials [8, 9] shows that as n, k such that k = O(n), (1) →∞ the top walk S (k) converges to FTW and the gap distribution “in bulk” converges to a discrete version of FSine. A similar asymptotic result was also obtained for domino tilings of an Aztec diamond [33]. Certain polynuclear growth models, last passage percolation problems and a bus system problem [4, 34, 42, 46] have also been analyzed in depth using nonintersecting path techniques. In each of the cases described above, the random walks are very specific and the analysis relies heavily on their particular properties. 1.2. Statement of theorems. Let k be a positive integer. Let 2i k (9) x = − , i 0,...,k . i k ∈ { } Note that x [ 1, 1] for all i. Let Y j k,Nk be a family of indepen- i ∈ − { l }j=0,l=1 dent identically distributed random variables where Nk is a positive integer. E j j Assume that Yl = 0 and Var(Yl ) = 1. Further, assume that there exists λY j λ > 0 such that E(e l ) < for all λ < λ . 0 ∞ | | 0 Define the random walk process S(t) = (S0(t),...,Sk(t)) by tNk/2 2 | | j tNk tNk j (10) Sj(t)= xj + Yi + Y tN /2 +1 sNk 2 − 2 | k | ! Xi=1 for t [0, 2], ∈ which starts at Sj(0) = xj. For Nk equally spaced times, Sj is given by 2 2 j j (11) Sj l = xj + (Y1 + + Yl ), l = 1, 2,...,Nk. Nk sNk · · · 2 2 For t between l and (l + 1), Sj(t) is simply defined by linear interpo- Nk Nk lation. Let (C([0, 2]; Rk+1), ) be the family of measurable spaces constructed from the continuous functionsC on [0, 2] taking values in Rk+1 equipped with 6 J. BAIK AND T. M. SUIDAN their Borel sigma algebras (generated by the sup norm). Let Ak,Bk be the events defined by ∈C (12) A = y (t) < Theorem 1 (Edge). Let Pk be the probability measure induced on (C([0, 2]; k+1 R ), ) by the random walks S(t) : t [0, 2] . Let hk k>0 and Nk k>0 satisfyC (14). Then { ∈ } { } { } (17) lim Pk(Ck Ak Bk)= FTW(ξ). k | ∩ →∞ A similar theorem holds for the bulk. Theorem 2 (Bulk). Let Pk be the probability measure induced on (C([0, 2]; k+1 R ), ) by the random walks S(t) : t [0, 2] and let hk k>0 and Nk k>0 satisfyC (14). Then { ∈ } { } { } (18) lim Pk(Dk Ak Bk)= FSine(η). k | ∩ →∞ NONINTERSECTING RANDOM WALKS 7 The proofs have a two-step strategy. The first step is to show that under the conditions of the theorems, the process S(t) is well approximated by nonintersecting Brownian bridge processes starting and ending at the same positions. This proof relies on the Komlos–Major–Tusnady (KMT) theorem. The second step is to compute the limiting distributions of the noninter- secting Brownian bridge processes and prove that these distributions are indeed FTW or FSine. This process is quite similar to the one discussed in Section 1.1.1, with the minor change that the Brownian bridge processes start and end at equally spaced locations, rather than at the same location. This change results in a Coulomb-gas density with the so-called Stieltjes– Wigert potential instead of the quadratic potential which appears in the GUE case. Such a nonintersecting Brownian bridge process was also con- sidered in [24, 26] and the connection to the Stieltjes–Wigert potential was made in [26] in order to compute the partition function and the limiting density of states. However, the edge and bulk scaling limits of the system had not been worked out. This paper obtains the asymptotics of the or- thogonal polynomials with respect to the Stieltjes–Wigert weight by using the Riemann–Hilbert method. As a consequence, the edge and bulk scaling limits are obtained. The above theorems are proved under the condition that Nk is large compared to k + 1, the number of particles. This assumption ensures that the Brownian approximation of the random walks has a smaller effect than the nonintersecting condition. Although it is believed that the condition on Nk is technical, it is not clear under which conditions on the random variables one has N = O(k). For example, when Y j are Bernoulli, these k { l } results were proven even when Nk = O(k) (see Section 1.1.3 above). This is because there is an integrability in this problem: the Karlin–McGregor argument applies directly because intersecting paths must be incident at some time. It is a challenge to find the optimal scaling such that a result of this nature holds for more general random variables. In other words, in what scaling regime does the exact Karlin–McGregor calculation essentially not matter? This paper is organized as follows. The approximation by the Brownian bridge process is proved in Section 2. The asymptotic analysis of the Brow- nian bridge process (appearing in Section 2) is carried out in Section 3. Some other considerations such as finite-dimensional distributions and the modifications necessary to study random variables without finite moment generating functions are discussed in Section 4. 2. Approximation by a Brownian bridge process. Let Xt t 0 be the k+1 { } ≥ R -valued stochastic process Xt = (X0(t),...,Xk(t)), where Xj(t)= xj + j j Bt for a family of k + 1 independent standard Brownian motions Bt . The 8 J. BAIK AND T. M. SUIDAN proof in this section relies on the Komlos–Major–Tusnady coupling of Brow- nian motions and random walks [38, 39] which can be stated in our set- j k,Nk ting as follows. With increments of the form Yl j=0,l=1 described in the Introduction, there exists a coupling such that{ } 2l 2l 1 ax (19) P sup Si Xi > (c log N + x) e− N − N √N k ≤ 0 l Nk k k k ≤ ≤ for some fixed a, c > 0 which depend only on the properties of the moment j k,Nk generating functions of the Yl j=0,l=1. Alternatively, (19) can be written as { } 2l 2l c log Nk ay√N (20) P sup S X > + y e− k . i N − i N √N ≤ 0 l Nk k k k ≤ ≤ This fact immediately implies that 2l 2l c log Nk P sup sup Si Xi > + y N − N √N 0 i k 0 l Nk k k k (21) ≤ ≤ ≤ ≤ ay√N (k + 1)e− k . ≤ Let S(t) t [0,2] be the (k + 1)-dimensional random walk process defined { } ∈ in the Introduction and let Xt be the KMT-coupled (k + 1)-dimensional Brownian process on the same{ probability} spaces (Ω(k), (k), P(k)). We can assume that the probability space which holds S and XFis large enough to hold a third process Zt = (Z0(t),...,Zk(t)), where the Zi(t) are standard Brownian bridge processes with initial and terminal conditions specified by Z (0) = Z (2) = x . Let F S, F X , F Z : C([0, 2], Rk+1) R be defined by i i i k k k → S ½Ak Bk Ck (y) (22) Fk (y)= ∩ ∩ , E½A B (S) k∩ k X ½Ak Bk Ck (y) (23) Fk (y)= ∩ ∩ , E½A B (X) k∩ k Z ½Ak Bk Ck (y) (24) Fk (y)= ∩ ∩ . E½A B (Z) k∩ k Let GS, GX , GZ : C([0, 2], Rk+1) R be defined by k k k → S ½Ak Bk Dk (y) (25) Gk (y)= ∩ ∩ , E½A B (S) k∩ k X ½Ak Bk Dk (y) (26) Gk (y)= ∩ ∩ , E½A B (X) k∩ k Z ½Ak Bk Dk (y) (27) Gk (y)= ∩ ∩ . E½A B (Z) k∩ k NONINTERSECTING RANDOM WALKS 9 Theorems 1 and 2 will be proven in two steps. The first step is to show that under the conditions given in the Introduction, the following holds. S Z S Z Proposition 1. As k , the random variables Fk , Fk , Gk and Gk satisfy →∞ (28) E(F S(S) F Z(Z)) 0, k − k → (29) E(GS (S) GZ (Z)) 0. k − k → E S P E S P As (Fk (S)) = k(Ck Ak Bk) and (Gk (S)) = k(Dk Ak Bk), it is enough to prove the following.| ∩ | ∩ Proposition 2. E Z (30) lim Fk (Z)= FTW(ξ), k →∞ E Z (31) lim Gk (Z)= FSine(η). k →∞ The proof of Proposition 2 is given in Section 3 below. The rest of this section focuses on the proof of Proposition 1. Proposition 1 is proved in two E S E X E X steps: first, (Fk (S)) is approximated by (Fk (X)) and second, (Fk (X)) E Z is approximated by (Fk (Z)). The proof of (29) is handled in a similar way. Three preliminary lemmas are needed in order to prove Proposition 1. Recall from (9) that 2i k (32) x = − , i = 0,...,k. i k Lemma 1. Let a, c > 0 be the constants in the KMT approximation (21). For any ρ 3c log Nk , ≥ √Nk (1/2)a√Nkρ ½ E ½ Ak Bk (S) Ak Bk (X) 2(k + 1)e− (33)| ∩ − ∩ |≤ 32(k + 1) Nk ρ2N /64 + e− k + 8(2k + 1)ρ, ρ s π (1/2)a√Nkρ ½ E ½ Bk (S) Bk (X) (k + 1)e− (34) | − |≤ 16(k + 1) Nk ρ2N /64 + e− k + 8(k + 1)ρ, ρ s π (1/2)a√Nkρ ½ E ½ Ck (S) Ck (X) (k + 1)e− (35) | − |≤ 16(k + 1) Nk ρ2N /64 + e− k + 8(k + 1)ρ, ρ s π 10 J. BAIK AND T. M. SUIDAN (1/2)a√Nkρ ½ E ½ Dk (S) Dk (X) (k + 1)e− (36) | − |≤ 16(k + 1) Nk ρ2N /64 + e− k + 8(k + 1)ρ. ρ s π Proof. Note that ½ E ½A B (S) A B (X) | k∩ k − k∩ k | ½ ½ ½ E ½ = Ak (S) Bk (S) AK (X) Bk (X) (37) | − | ½ ½ ½ ½ ½ = E ( ½ (S) (X)) (S) + ( (S) (X)) (X) | Ak − Ak Bk Bk − Bk Ak | ½ ½ ½ E ½ (S) (X) + E (S) (X) . ≤ | Ak − Ak | | Bk − Bk | ½ E ½ P We first estimate Ak (S) Ak (X) = ( ), where = S Ak, X / Ak S / A , X A .| Recall− that A =| y (tE) < 2l 2l c log Nk ρ P( 1) P sup sup Si Xi > + E ≤ N − N √N 2 0 i k 0 l Nk k k k ≤ ≤ ≤ ≤ 2l 2l c log Nk ρ + P sup sup Si Xi + N − N ≤ √N 2 0 i k 0 l Nk k k k ≤ ≤ ≤ ≤ ρ max Xi(t) Xi(s) > ∩ s,t (2l/N ,2(l+1)/N ) | − | 2 ∈ k k (38) for some 0 i k and for some 0 l < Nk ≤ ≤ ≤ (1/2)a√N ρ (k + 1)e− k ≤ ρ + (k + 1)NkP max X1(t) X1(s) > t,s [0,2/N ] | − | 2 ∈ k NONINTERSECTING RANDOM WALKS 11 (1/2)a√N ρ 16(k + 1) Nk ρ2N /64 (k + 1)e− k + e− k . ≤ ρ s π Note that this estimate does not use the fact that is a subset of . For a E1 E path in the event 2, there exists i 1, 2,...,k such that mint [0,2](Si(t) E ∈ { } ∈ − Si 1(t)) < 0, but Xi 1(t) < Xi(t) for all t [0, 2] and Sj(t) Xj(t) − − c log Nk ∈ | − |≤ + ρ all t [0, 2] and j 0, 1,...,Nk . Therefore, for a path in 2, √Nk ∈ ∈ { } E 2c log Nk 0 < mint [0,2](Xi(t) Xi 1(t)) < + 2ρ 4ρ. Thus, from a standard ∈ − − √Nk ≤ Brownian motion argument, P( 2) P 0 min (Xi(t) Xi 1(t)) < 4ρ for some 1 i k E ≤ ≤ t [0,2] − − ≤ ≤ (39) ∈ kP 0 min (X1(t) X0(t)) < 4ρ 4kρ. ≤ ≤ t [0,2] − ≤ ∈ A similar argument yields that P( 3) P 4ρ< min (Xi(t) Xi 1(t)) < 0 for some 1 i k E ≤ − t [0,2] − − ≤ ≤ (40) ∈ kP 4ρ< min (X1(t) X0(t)) < 0 4kρ. ≤ − t [0,2] − ≤ ∈ Therefore, ½ E ½ (S) (X) | Ak − Ak | (41) = P( )+ P( )+ P( ) E1 E2 E3 (1/2)a√N ρ 16(k + 1) Nk ρ2N /64 (k + 1)e− k + e− k + 8kρ. ≤ ρ s π ½ E ½ P We now estimate Bk (S) Bk (X) = ( ), where = S Bk, X / B S / B , X B | . As before,− we express| F as F { ,∈ a disjoint∈ k} ∪ { ∈ k ∈ k} F F1 ∪ F2 ∪ F3 union. The first event, 1, is the subset of consisting of the same bad paths as in . The event Fis the intersectionF of and S B , X / B E1 F2 F \ F1 { ∈ k ∈ k} and the event 3 is the intersection of 1 and S / Bk, X Bk . The argument for F implies that the same boundF \ F (38) applies{ ∈ to P∈( ).} For a E1 F1 path in 2, there exists i 0, 1,...,k such that Xi(2) / [xi hk,xi + hk]. F ∈ { } 2∈ log Nk− But as Si(2) [xi hk,xi + hk] and Si(2) Xi(2) + ρ 2ρ, we ∈ − | − |≤ √Nk ≤ find that Xi(2) (xi + hk,xi + hk + 2ρ] or Xi(2) [xi hk 2ρ, xi hk). Therefore, ∈ ∈ − − − (42) P( ) (k + 1)P(X (2) [ 2ρ, 2ρ]) 4(k + 1)ρ. F2 ≤ 0 ∈ − ≤ A similar argument yields the same bound for P( 3). Therefore, (34) is proved, as is (33), by using (37) and (41). An almostF identical argument proves (35) and (36). 12 J. BAIK AND T. M. SUIDAN 2 Denote by p (a, b)= 1 e (a b) /(2t) the standard heat kernel in one t √2πt − − dimension. The theorem of Karlin and McGregor [36] for nonintersecting Brownian motions implies that the joint probability density function ft(y0,...,yk) of (k + 1)-dimensional Brownian motion X(t) at time t sat- isfying X (s) < X (s) < < X (s) for s [0,t] is equal to 0 1 · · · k ∈ k (43) ft(y0,...,yk) = det(pt(xi,yj))i,j=0, where xi = Xi(0). The following lemma establishes a lower bound for this density when yi = xi for all i. Lemma 2. For t> 0, k(k+1) k 1 2(k+1)(k+2)/(3tk) 2 (44) det(p (x ,x )) e− . t i j i,j=0 (k+1)/2 √ ≥ (2πt) tk In particular, for all sufficiently large k, k k2 (45) det(p (x ,x )) k− . 2 i j i,j=0 ≥ 2i k Proof. As xi = k− , we have k 2 k 1 1/(2t)(xi xj ) det(pt(xi,xj))i,j=0 = det e− − √2πt i,j=0 (46) k 2 j2 e− j=0 2 = det(e2ij/(tk ))k . (2πtP)(k+1)/2 i,j=0 It is an exercise to show that for k 1, ≥ k k 2ij/(tk2) k l(l 1)/2 j k+1 j (47) det(e )i,j=0 = δ − (δ 1) − , " #" − # lY=1 jY=1 2 where δ = e4/(tk ). Using (47) and the fact that δ 1 > 4 > 0, we obtain − tk2 k det(pt(xi,xj))i,j=0 k 1 2(k+1)(k+2)/(3tk) j k+1 j = (k+1)/2 e− (δ 1) − (2πt) j=1 − (48) Y 1 2(k+1)(k+2)/(3tk) k(k+1)/2 e− (δ 1) ≥ (2πt)(k+1)/2 − k(k+1)/2 1 2(k+1)(k+2)/(3tk) 4 (k+1)/2 e− 2 . ≥ (2πt) tk This completes the proof of Lemma 2. NONINTERSECTING RANDOM WALKS 13 The following lemma will be used to control the difference between a conditioned version of the process X and the process Z. 2 Lemma 3. If h (2k) 2k , then for sufficiently large k, k ≤ − hk hk det(p1(yi,xj)) h h det(p1(yi,xj + sj)) ds0 dsk 1 (49) − k · · · − k · · · , det(p (x ,x )) − hk hk ≤ k 2 i j R h R h det(p2(xi,xj + sj)) ds0 dsk − k · · · − k · · · uniformly in (y ,...,y )R Rk+1R. 0 k ∈ Proof. The conclusion of this lemma is a consequence of several ele- k mentary determinant estimates. First, note that if A = (aij)i,j=0 is a (k + ij 1) (k + 1) matrix with entries aij 1, then for the matrix I given by ij× | |≤ (I )mn = δimδjn, we have (50) det(A) det(A + εIij) εk!. | − |≤ Using a Lipschitz estimate for the Gaussian density, equation (50) implies that for any t 1 , any h> 0 and any (a , . . ., a ), (b , . . ., b ) Rk+1, ≥ √2πe 0 k 0 k ∈ 1 h h det(p (a , b )) det(p (a + s , b )) ds ds t i j − (2h)k+1 · · · t i i j 0 · · · k Z h Z h (51) − − 2h(k + 1)2k!. ≤ A simple algebraic manipulation now yields that the left-hand side of (49) equals det(p (y ,x )) (52) 1 i j Q2 + Q1 , det(p2(xi,xj)) · det(p2(xi,xj)) + 2 det(p2(xi,xj)) + 2 Q Q where 1 1 = [det(p1(yi,xj)) k+1 k+1 Q (2hk) [ hk,hk] (53) Z − det(p (y ,x + s ))] ds ds , − 1 i j j 0 · · · k 1 2 = [det(p2(xi,xj + sj)) k+1 k+1 Q (2hk) [ hk,hk] (54) Z − det(p (x ,x ))] ds ds . − 2 i j 0 · · · k Using the estimates (45) and (51), we obtain k2 2 1 k2 (55) det(p (x ,x )) + k− 2h (k + 1) k! k− 2 i j Q2 ≥ − k ≥ 2 for sufficiently large k. Hence, again using (51), 1 2 k2 (56) Q 2hk(k + 1) k!k . det(p2(xi,xj)) + 2 ≤ Q 14 J. BAIK AND T. M. SUIDAN On the other hand, as det(p (x ,y )) is the density function for (y ,...,y ) 1 i j 0 k ∈ Rk+1, where Rk+1 = (y ,...,y ) Rk+1 : y < k 2 1 (1/2)(xi yi) (57) det(p1(xi,yj)) e− − 1 ≤ √2π ≤ iY=0 and hence det(p1(yi,xj)) 2 2 2k2 (58) Q 2hk(k + 1) k!k . det(p2(xi,xj)) · det(p2(xi,xj)) + 2 ≤ Q 2k2 Since hk is assumed to be less than or equal to (2k)− , (49) follows. Proof of Proposition 1. Two estimates will be needed. Note that (59) E(F S(S) F Z (Z)) E F S(S) F X (X) + E(F X (X) F Z (Z)) . | k − k |≤ | k − k | | k − k | The first term on the right-hand side of (59) is estimated as follows: E F S(S) F X (X) | k − k | ½ ½A B C (S) A B C (X) = E k∩ k∩ k k∩ k∩ k ½ E½A B (S) E (X) k k − Ak Bk ∩ ∩ ½ ½ = E (½A B C (S)(E A B (X) E A B (S)) (60) | k∩ k∩ k k∩ k − k∩ k ½ ½ + ( ½A B C (S) A B C (X))E A B (S)) k∩ k∩ k − k∩ k∩ k k∩ k 1 ½ (E½A B (S)E A B (X))− × k∩ k k∩ k | ½ ½ ½ E½A B (X) E A B (S) ( A B C (S) A B C (X)) | k∩ k − k∩ k | + E k∩ k∩ k − k∩ k∩ k ½ E½ (X) E (X) ≤ Ak Bk Ak Bk ∩ ∩ ½ ½ ½ 2E ½A B (S) A B (X) + E C (S) C (X) | k∩ k − k∩ k | | k − k |. ≤ E½A B (X) k∩ k 1/4 By setting ρ = Nk− in Lemma 1, for sufficiently large k, it is easy to check that 20k ½ E ½A B (S) A B (X) , | k∩ k − k∩ k |≤ 1/4 Nk (61) 20k ½ E ½ (S) (X) . | Ck − Ck |≤ 1/4 Nk NONINTERSECTING RANDOM WALKS 15 On the other hand, by using (45) and the argument leading to (55), for sufficiently large k, E½A B (X) k∩ k k+1 = (2hk) det(p2(xi,xj)) k+1 E½ + ( Ak Bk (X) (2hk) det(p2(xi,xj))) (62) ∩ − k+1 = (2hk) det(p2(xi,xj)) + (det(p2(xi,xj + sj)) det(p2(xi,xj))) ds0 dsk [ h ,h ]k+1 − · · · Z − k k (2h )k+1 k . ≥ 2kk2 Hence, from (61), for sufficiently large k, k2+1 E S X 120k (63) Fk (S) Fk (X) 0 | − |≤ k+1 1/4 → (2hk) Nk as k . For the second term of (59), note that the Karlin–McGregor for- mula→∞ for nonintersecting Brownian motions implies that [cf. (43) above] the density function of the nonintersecting Brownian bridge process Z evaluated at time 1 is equal to k k det(p1(xi,yj))i,j=0 det(p1(yi,xj))i,j=0 (64) f(y0,...,yk)= k . det(p2(xi,xj))i,j=0 Similarly, the density of the nonintersecting Brownian motion X evaluated at time t is equal to f(y0,...,yk) (65) k k [ h ,h ]k+1 det(p1(xi,yj))i,j=0 det(p1(yi,xj + sj))i,j=0 ds0 dsk = − k k · · · . R [ h ,h ]k+1 det(p2(xi,xj + sj)) ds0 dsk − k k · · · Therefore, R E(F X (X) F Z (Z)) | k − k | ½ ½A B C (X) A B C (Z) = E k∩ k∩ k k∩ k∩ k ½ E½ (X) − E (Z) Ak Bk Ak Bk ∩ ∩ det(p1(xi,yj)) ≤ Rk+1 k+1 Z > Z[ hk,hk] (66) − det(p1(yi,xj + sj)) ds0 dsk × · · · 16 J. BAIK AND T. M. SUIDAN 1 − det(p2(xi,xj + sj)) ds0 dsk × [ h ,h ]k+1 · · · Z − k k det(p1(xi,yj)) det(p1(yi,xj)) dy0 dyk. − det(p2(xi,xj)) · · · By using Lemma 3 and (57), this implies that E X Z 1 (Fk (X) Fk (Z)) det(p1(xi,yj)) dy0 dyk | − |≤ k Rk+1 | | · · · Z > (67) k 2 1 1 (xi yi) /2 1 e− − dy0 dyk . Rk+1 ≤ k > " √2π # · · · ≤ k Z iY=0 The proof of (29) is exactly the same. This completes the proof of Proposi- tion 1. 3. Asymptotics of a Brownian bridge process. We prove Proposition 2 in this section. Together with the results of Section 2, this completes the proofs of Theorem 1 and Theorem 2. From the density formula of Karlin and McGregor for a nonintersecting Brownian bridge processes [36] [cf. (43)], E Z 1 (Fk (Z)) = k det(p2(xi,xj))i,j=0 (68) k k 2 [det(p1(xi,yj))i,j=0] (1 1(yj)) dyj, Rk+1 × > −H Z jY=0 2i k where xi = k− and (69) 1(y)= ½(√2k+ξ/(√2k1/6), )(y). H ∞ Also, E Z 1 (Gk (Z)) = k det(p2(xi,xj))i,j=0 (70) k k 2 [det(p1(xi,yj))i,j=0] (1 2(yj)) dyj, Rk+1 × > −H Z jY=0 where (71) 2(y)= ½[ η/√k+1,η/√k+1](y). H − We need the limit of (68) and (70) as k . →∞ NONINTERSECTING RANDOM WALKS 17 In the discussion below, (y) denotes either 1 or 2. Indeed, the alge- bra below works for arbitraryH bounded functionsH (yH). Using the formula H for pt and the definition of xi, an elementary algebraic manipulation using Vandermode determinants yields that (68) and (70) are equal to k 2 2yj /k 2yi/k 2 yj 2yj (72) Ck′ (e e ) (1 (yj))e− − dyj, · Rk+1 − −H Z 0 i 1 (k2/4)(log u)2 (k2/4)(log u)2 log u (77) w(u)= e− = e− − . u m m Note that w(u)= o(u− ) for any m 0 as u + , and w(u)= o(u ) for any m 0 as u 0. ≥ → ∞ ≥ ↓ 2/k2 With the change of variables u = e− x, (77) becomes (k2/4)(log x)2 1/k2 (78) w(u) du = c e− dx, c = e . · This is, up to a constant, the Stieltjes–Wigert weight, which is defined as 1/2 k2(log x)2 (79) π− ke− 18 J. BAIK AND T. M. SUIDAN (see, e.g., Section 2.7 of [51] or Section 3.27 of [37]). The moments for the Stieltjes–Wigert weight is an example of an indeterminate moment prob- lem; hence, there are several weights that have the same moments as the weight (79). Another interesting feature of the Stieltjes–Wigert weight (79) is that the corresponding orthogonal polynomials (called Stieltjes–Wigert 1/(2k2) polynomials) are examples of so-called q-polynomials with q = e− (see, e.g., Section 3.27 of [37]). The connection between the nonintersecting Brow- nian bridge process Z and the Stieltjes–Wigert weight was first observed in [26]; the Stieltjes–Wigert weight also appears in [28]. Various β = 2 matrix ensembles of the form (74) (on both the real line and subsets of the real line) have been analyzed asymptotically and it has been proven that the local statistics of the “eigenvalues” (or the particles u0, . . ., uk) are generically independent of the potential w. For example, such (k+1)V (x) “universality” is proved when w(x)= e− for an analytic weight V on R or R+ satisfying certain growth conditions as x (and as x 0 →±∞ Q(x) → for weights on R+) (e.g., [11, 21, 40, 44]) and when w(x)= e− , where Q(x) is a polynomial (e.g., [20]). However, the asymptotic analysis of the ensemble with the weight given in (77) above does not seem to appear in the literature. It is well known [see (80) below] that the asymptotics of β = 2 ensembles amount to the asymptotic analysis of the corresponding orthogonal polynomials. For our case, we need the asymptotics of the or- thogonal polynomials of degree k and k + 1 with respect to the weight (77) as k ; note that the weight also varies as k increases. The asymptotics of Stieltjes–Wigert→∞ polynomials were recently studied in [29] and [55], but in different asymptotic regimes: the degree goes to infinity while the weight is fixed. Therefore, the analysis of this section seems to yield new results for asymptotics of Stieltjes–Wigert polynomials. Nevertheless, the asymptotic analysis of the orthogonal polynomials and the ensemble (74) with varying weight (77) can be done in a very similar way to the analysis in [20, 21] using the Deift–Zhou steepest-descent method for related Riemann–Hilbert prob- lems (RHP’s), which is now one of standard tools for asymptotic analysis of orthogonal polynomials. We note that [55] also used the Deift–Zhou method (for a different asymptotic regime) and our analysis has some overlap with the analysis of [55]. In this section, we present only a sketch of the analysis. It is a standard result in random matrix theory (see, e.g., [41, 52]) that (74) equals (80) det(1 K ˆ), − kH where γ p (x)p (y) p (x)p (y) (81) K (x,y)= w(x)w(y) k k+1 k − k k+1 k γ x y q k+1 − NONINTERSECTING RANDOM WALKS 19 n is the Christoffel–Darboux kernel in which pn(x)= γnx + is the nth orthonormal polynomial with respect to w. Hence, · · · E Z ˆ (Fk (Z)) = det(1 KkH1), (82) − E(GZ (Z)) = det(1 K Hˆ ). k − k 2 Let Y(z) be the solution to the following Riemann–Hilbert problem: Y(x) is the 2 2-matrix-valued function on C R satisfying × \ + Y(z) is analytic for z C R+, Y (z) = limε 0 Y(z iε) is continuous for • ∈ \ ± ↓ ± z R+ and Y(z) is bounded as z 0; for∈ z R , → • ∈ + 1 w(z) (83) Y (z)= Y (z) ; + 0 1 − (k+1)σ3 1 Y(z)z− = (I + O(z− )) uniformly as z such that z C R+, • 1 0 →∞ ∈ \ where σ3 = 0 1 . − There is a unique solution Y to this RHP and, in particular, the (11) 1 and (21) entries of Y(z) are given by Y11(z)= γk−+1pk+1(z) and Y21(z)= 2πiγkpk(z) [25]. Note that the existence of Y under the condition that − 1 Y(z) is bounded as z 0 (rather than, e.g., that Y12(z)= O(z− ), as in, say, [54]) is due to the→ fact that w(x) 0 faster than any polynomial as x 0. Thus, the Christoffel–Darboux kernel→ can be written as, by using the fact→ that det Y(z)=1, 1 1 1 (84) K (x,y)= w(x)w(y) (0 1) Y− (y)Y(x) . k 2πi(x y) 0 q − One of the main ingredient in analyzing the RHP for orthogonal poly- nomials asymptotically is the so-called equilibrium measure and the corre- sponding “g-function.” Let ψ(x) dx be a measure on R+ = supp(w) with total mass (85) ψ(x) dx = k + 1. Z Define the “G-function” (86) G(z)= log(z x)ψ(x) dx, z C R , − ∈ \ + Z where log represents the log function on the standard branch so that log u = log u + i arg(u), where arg(u) <π. It is customary to define ψ to be the probability| | measure and| define| the g-function as in (86) [hence, G(z)= (k +1)g(z)], but in this paper, we use the above convention since it simplifies some formulas below. Note that (87) G+(x)+ G (x) = 2 log x y ψ(y) dy, x R+. − | − | ∈ Z 20 J. BAIK AND T. M. SUIDAN We look for G satisfying the following two conditions: there exists a constant ℓ such that G+(x)+ G (x) + log(w(x)) ℓ =0 for x supp(ψ), • − − ∈ G+(x)+ G (x) + log(w(x)) ℓ< 0 for x R+ supp(ψ). • − − ∈ \ For such G, the measure ψ is called the equilibrium measure. Using the standard procedure to solve this variational problem (see, e.g., [19, 47]), one can compute the equilibrium measure for the weight (77). Lemma 4. For the weight (77), the support of the equilibrium measure is [a, b], where 2 √a = e(2k+1)/k e(4k+2)/k2 e2/k, − − (88) q 2 √b = e(2k+1)/k + e(4k+2)/k2 e2/k. − The equilibrium measure is, for x [aq, b], ∈ 1 1 (log(w(s))) ψ(x)= (b x)(x a)h(x), h(z)= − ′ ds, 2π − − 2πi (s z)R(s) IC (89) q − where R(z) = ((z a)(z b))1/2 denotes the principal branch of the square root function and− the simple− closed contour C contains z and [a, b], inside does not touch ( , 0] and is oriented counterclockwise. A residue calcula- tion yields that −∞ k2 (b x)(x a) (90) ψ(x)= arctan − − , x [a, b]. 2πx √ p ab + x ∈ We remark that a and b are sometimes called the Mhaskar–Rakhmanov– Saff numbers. The above a and b are obtained in [55]: with αn and βn as in (2.2) and (2.3) of [55], we have 1/(2k2) 1/(2k2) (91) a = (e− αn) k k/2,n=k+1, b = (e− βn) k k/2,n=k+1. | 7→ | 7→ Given this ψ, G(z) is defined as in (86) and ℓ is defined as ℓ = 2G(b) log(w(b)) = 2G(a) log(w(a)). The function h(z) in (89) is analytic in z − C ( , 0]. A residue− calculation yields that ∈ \ −∞ k2 √ab + z R(z) (92) h(z)= log − , z C ( , 0], 2zR(z) √ ab + z + R(z) ∈ \ −∞ where log denotes the principal branch of logarithm. For a computation below, we note that as k , →∞ 2 2 3/2 √a = 1 + + O(k− ), − k k (93) r 2 2 3/2 √b =1+ + + O(k− ). rk k NONINTERSECTING RANDOM WALKS 21 We also remark that with x =1+ 2w , for w = O(1), as k , at least √k →∞ formally, k (94) ψ(x) dx 2 w2 dw, w [ √2, √2], ∼ π − ∈ − p which is precisely Wigner’s semicircle. This last calculation is not going to be used below, but it provides an intuitive reason as to why the ensemble (74) [and (72)] has the same asymptotics as the Gaussian unitary ensemble, not only locally, but also globally. Set (1/2)ℓσ3 G(z)σ3 (1/2)ℓσ3 (95) M(z)= e− Y(z)e− e for z C R+. Using the analyticity of G for z R+ [a, b] and the varia- tional∈ conditions,\ M(z) solves the following, equivalent,∈ \ RHP: M(z) is analytic for z C R+, M (z) is continuous for z R+ and M(z) • is bounded as z 0; ∈ \ ± ∈ → for z R+, M+(z)= M (z)VM (z), where • ∈ − eG−(z) G+(z) 1 (96) V (z)= − , z (a, b), M 0 eG+(z) G−(z) − ∈ 1 e2G(z)+log(w(z)) ℓ (97) V (z)= − , z R (a, b); M 0 1 + ∈ \ M(z)= I + O(z 1) as z . • − →∞ The nonunit terms in the jump matrix can be expressed in a unifying way. Set (98) H(z)= G(z)+ 1 log(w(z)) 1 ℓ, z C (( , 0] [a, b]). 2 − 2 ∈ \ −∞ ∪ Noting the variational condition, we find that for z (a, b), ∈ G+(z) G (z) = 2G+(z) + log(w(z)) ℓ = 2H+(z) (99) − − − = (2G (z) + log(w(z)) ℓ)= 2H (z). − − − − − Hence, G+(z) G (z) has an analytic continuation both above and the − − below the real axis. Therefore, the jump matrix VM equals e 2H+(z) 1 (100) V (z)= − , z (a, b), M 0 e 2H−(z) − ∈ 1 e2H(z) (101) V (z)= , z R (a, b). M 0 1 + ∈ \ Using the definition of G and Lemma 4, one can check that 1 (102) H′(z)= 2 R(z)h(z). 22 J. BAIK AND T. M. SUIDAN We now scale the RHP for M so that the interval (a, b) becomes ( 1, 1). In other words, instead of moving the interval as the support of the equilib-− rium measure, we will fix the support. In that way, we can use the analysis of [20, 21] more directly. Define b a b + a (103) N(z)= M − z + . 2 2 b+a Set Σ = ( b a , ) and set − − ∞ b a b + a (104) Hˆ (z)= H − z + . 2 2 The matrix N solves the following RHP: N(z) is analytic for z C Σ, N (z) is continuous for z Σ and N(z) is • b+a∈ \ ± ∈ bounded as z b a ; →− − for z Σ, N+(z)= N (z)VN (z), where • ∈ − 2Hˆ+(z) (105) V (z)= e− 1 , z ( 1, 1), N 2Hˆ−(z) 0 e− ∈ − 1 e2Hˆ (z) (106) VN (z)= , z Σ ( 1, 1); 0 1 ∈ \ − N(z)= I + O(z 1) as z . • − →∞ Note the factorization for z ( 1, 1), ∈ − 2Hˆ+(z) e− 1 1 0 0 1 1 0 ˆ = ˆ ˆ , 2H−(z) e 2H−(z) 1 1 0 e 2H+(z) 1 0 e− − − (107) − where we use the fact that Hˆ+(z)+ Hˆ (z)=0 for z ( 1, 1). Let Σj, j = − ∈ − 0, 1,..., 4, and Ωj, j = 1,..., 4, be the contours and open regions given in Figure 1. Contours are oriented from left to right. Define N(z), z Ω1 Ω4, 1 0 ∈ ∪ N(z) ˆ , z Ω , (108) Q(z)= e 2H(z) 1 2 − ∈ − 1 0 N(z) ˆ , z Ω . e 2H(z) 1 3 − ∈ Then Q+(z)= Q (z)VQ(z) for z in Σ0,..., Σ4, where − 0 1 (109) V (z)= , z Σ , Q 1 0 ∈ 0 − 1 0 (110) V (z)= ˆ , z Σ Σ , Q e 2H(z) 1 1 2 − ∈ ∪ 1 e2Hˆ (z) (111) VQ(z)= , z Σ3 Σ4. 0 1 ∈ ∪ NONINTERSECTING RANDOM WALKS 23 Fig. 1. Contours for N. The off-diagonal terms of VQ on Σ1 Σ4 converge to 0 as the following lemma implies. ∪···∪ Lemma 5. There exist δ > 0 and k > 0 such that for k k , 0 0 ≥ 0 (112) Re[Hˆ (x + iy)] 2k y 1 x2 for 1 x 1 and δ y δ . ≥ | | − − ≤ ≤ − 0 ≤ ≤ 0 For any δ> 0, p b + a (113) Hˆ (x) kδ3/2 for Hence, VQ V for a constant matrix V defined as → ∞ ∞ 0 1 (115) V (z)= , z Σ0, ∞ 1 0 ∈ − and V (z)= I for z Σ1 Σ4, where the convergence VQ V is in ∞ ∈ ∪···∪2 → ∞ L∞(Σ0 Σ4) and also in L ((Σ0 Σ4) z 1 > δ z +1 > δ ) for an arbitrary,∪···∪ but fixed, δ> 0. Let∪···∪ ∩{| − | }∩{| | } z 1 1/4 (116) β(z)= − , z + 1 where the branch cut is [ 1, 1] and β(z) 1 as z + on the real line, and define − ∼ → ∞ β + β 1 i(β β 1) (117) Q (z)= 1 − − ∞ 2 i(β β 1) − β +−β 1 − − − C for z Σ0. Then Q∞(z) is the solution to the RHP for the Q+∞ = Q∞V ∈ \ − ∞ and Q∞(z) I as z . The convergence VQ V is not uniform near → →∞ → ∞ 24 J. BAIK AND T. M. SUIDAN the points z = 1, hence it is not true that Q(z) Q∞(z) for all z and one therefore needs± local parametrix for z in a neighborhood→ of 1. Let Ψ(z) be the matrix-valued function constructed from± the Airy func- tion and its derivatives, as defined in Proposition 7.3 of [20]. Let ε> 0. For z U := z : z 1 < ε , set ∈ r { | − | } 3 2/3 Hˆ (z)σ3 (118) S (z)= E(z)Ψ(( Hˆ (z)) )e− , r − 2 where 1 1 E(z)= √πe(π/6)i i −i (119) − − 3 ˆ 1/6 1 ( 2 H(z)) β(z)− 0 − 3 1/6 . × 0 ( Hˆ (z))− β(z) ! − 2 Note that E(z) is analytic in Ur if ε is chosen sufficiently small. The matrix S (z) is defined in a similar way for z U := z : z + 1 < ε . Define l ∈ l { | | } Q∞(z), z C Ur Ul Σ, (120) Q (z)= S (z), z ∈ U \ Σ,∪ ∪ par r r S (z), z ∈ U \Σ. l ∈ l \ From the basic theory of RHP, the estimate in Lemma 5 and the same 1 argument as in [20], one can check that the jump matrix for Qpar− Q converges to the identity in L2 L . Hence, ∩ ∞ 1 (121) Q(z) = (I + O(k− ))Qpar(z). This holds uniformly for z outside an open neighborhood of the contours Σ ∂Ur ∂Ul. But a simple deformation argument implies that the result is∪ extended∪ to z on the contours (see [20]). Hence, by reversing the trans- formations Y M N Q [see (95), (103) and (108)], the asymptotics of Y(z) for all→z C→are obtained.→ By substituting∈ the asymptotics of Y into (84), edge and bulk scaling limits of the Kk are obtained; see [15, 18, 21] for details. For x0 such that √k(x0 1) lies in a compact subset of (√k(a 1), √k(b 1)), for all ξ, η in a compact− subset of R, − − 1 ξ η (122) Kk x0 + ,x + S(ξ, η) ψ(x0) ψ(x) ψ(x0) → in trace norm for ξ, η R, where ∈ sin(π(ξ η)) (123) S(ξ, η)= − . π(ξ η) − 1 Here, we may replace ψ(x0) by Kk(x0,x0). The error is O(k− ), uniformly for ξ, η in a compact set. The convergence is also in trace norm in the NONINTERSECTING RANDOM WALKS 25 2 2/k Hilbert space L (( η, η)) for a fixed η> 0. From (82), by taking x0 = e , the limit (30) in Proposition− 2 is obtained. At the edge of the support of ψ(x), set 1 2/3 k7/6 (124) B = √b ah(b) . k 2 √ − − ∼ 2 As k , →∞ 1 ξ η (125) Kk b + , b + A(ξ, η) Bk Bk Bk → in trace norm in the Hilbert space L2((ξ, )) for a fixed ξ, where ∞ Ai(ξ)Ai′(η) Ai′(ξ)Ai(η) (126) A(ξ, η)= − ξ η − is the Airy kernel. Hence, from (82), the limit (31) in Proposition 2 is ob- tained. 4. Generalizations and discussions. We comment on three issues in this section: the case in which the moment generating function does not ex- ist, finite-dimensional distributions and the connection of this work to q- orthogonal polynomials. No moment generating function. In this paper, we have assumed the existence of the moment generating function for the random variable in- crements of nonintersecting random walks. This is simply to improve the E j 2+δ estimates. For the case Xi < , δ> 0, there is a version of the KMT theorem which provides analogous| | estimates∞ to those used in Section 2. As one would expect for this case, Nk must grow more rapidly in k. Another method for achieving results similar to those of this paper is to use Skorohod embedding in order to embed the nonintersecting random walks into Brow- nian motions. In order to achieve this, one must assume that E Xj 4 < . | i | ∞ Finite-dimensional distributions. The results of this paper focus on the limiting distributions of nonintersecting random walks at the fixed time t = 1. It is also interesting to consider finite-dimensional distributions of the 1/3 1/3 process, that is, in the correct scaling t1,...,tn [1 Ak− , 1+ Ak− ], the finite-dimensional distributions of the fluctuations∈ − of the top random walk should converge to those of the Airy process. A similar, but differently scaled, result should also be true “in bulk”; see, for example, [1, 46, 53] and references therein concerning the Airy process and other processes from random matrix theory. The methods of Section 2 are certainly applicable to this problem, however, the convergence of the finite-dimensional distri- butions of the nonintersecting Brownian bridges to Airy/sine processes does 26 J. BAIK AND T. M. SUIDAN not follow immediately from the analysis of Section 3. However, one can use a different approach based on the method of Eynard and Mehta [14, 23]. 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