Maximal height of non-intersecting Brownian motions

Maximal height of non-intersecting Brownian motions G. Schehr Laboratoire de Physique Théorique et Modèles Statistiques CNRS-Université Paris Sud-XI, Orsay Collaborators: A. Comtet (LPTMS, Orsay) P. J. Forrester (Dept. of Math., Melbourne) S. N. Majumdar (LPTMS, Orsay) J. Rambeau (Inst. Theor. Phys., Cologne) J. Randon-Furling (Univ. Paris 1, Paris) G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 1 / 44

Maximal height of non-intersecting Brownian motions G. Schehr Laboratoire de Physique Théorique et Modèles Statistiques CNRS-Université Paris Sud-XI, Orsay References: G. S., S. N. Majumdar, A. Comtet, J. Randon-Furling, Phys. Rev. Lett. 101, 150601 (2008) J. Rambeau, G. S., Europhys. Lett. 91, 60006 (2010); Phys. Rev. E 83, 061146 (2011) P. J. Forrester, S. N. Majumdar, G. S., Nucl. Phys. B 844, 500 (2011) G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 1 / 44

Height of rooted planar tree Height of rooted plane trees H 1,n with n + 1 nodes 0 1 2 3 H 1,n=3 = 3 H 1,n=3 = 2 H 1,n=3 = 2 H 1,n=3 = 1 Q : H 1,n for large n? G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 2 / 44

Height of rooted planar tree Mapping between rooted plane trees and Dyck paths 3 2 1 0 2n = 6 steps H 1,n H 1,n π lim = H 1 = n 2n 2 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 2 / 44

Generalization to N non-intersecting Dyck paths H N=8,n 2n steps G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 3 / 44

Generalization to N non-intersecting Dyck paths H N=8,n 2n steps = numerical estimate for H N=8,n G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 3 / 44

Generalization to N non-intersecting Dyck paths H N,n num 2n = H N num 1.67 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 4 / 44

Generalization to N non-intersecting Dyck paths H N,n num 2n = H N num 1.67 N H N,n num 2n = H N num 0.82 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 4 / 44

Generalization to N non-intersecting Dyck paths H N,n num 2n = H N num 1.67 N H N,n num 2n = H N num 0.82 N Q : can one compute H N? G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 4 / 44

Non-intersecting Brownian motions in 1d N Brownian motions in one-dimension ẋ i (t) = ζ i (t), ζ i (t)ζ j (t ) = δ i,j δ(t t ) x 1 (0) < x 2 (0) <... < x N (0) Non-intersecting condition x i (t) x 4 (0) x 1 (t) < x 2 (t) <... < x N (t), t 0 x 3 (0) x 2 (0) x 1(0) 0 t G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 5 / 44

Non-intersecting Brownian motions in 1d N Brownian motions in one-dimension ẋ i (t) = ζ i (t), ζ i (t)ζ j (t ) = δ i,j δ(t t ) x 1 (0) < x 2 (0) <... < x N (0) Non-intersecting condition x i (t) N = 4 x 1 (t) < x 2 (t) <... < x N (t), t 0 0 1 watermelons G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 5 / 44

Non-intersecting Brownian motions in 1d N Brownian motions in one-dimension ẋ i (t) = ζ i (t), ζ i (t)ζ j (t ) = δ i,j δ(t t ) x 1 (0) < x 2 (0) <... < x N (0) Non-intersecting condition x i (t) N = 4 x 1 (t) < x 2 (t) <... < x N (t), t 0 0 1 t watermelons "with a wall" G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 5 / 44

Vicious walkers in physics G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 6 / 44

Vicious walkers in physics and maths P. G. de Gennes, Soluble Models for fibrous structures with steric constraints (1968) M. E. Fisher, Walks, Walls, Wetting and Melting (1984) D. J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices (1999) C. Krattenthaler, A. J. Guttmann, X. G. Viennot, Vicious walkers, friendly walkers and Young tableaux (2000) P. L. Ferrari, K. Johansson, N. O Connell, M. Praehofer, H. Spohn, C. Tracy, H. Widom... Stochastic growth models, directed polymers (from 2000)... G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 6 / 44

Extreme statistics of vicious walkers x i (t) Maximal height of watermelons H 4 x 1 (t) < x 2 (t) <... < x N (t) 0 1 t H N = max[x N (τ), 0 τ 1] τ H N =? G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 7 / 44

Extreme statistics of Brownian motion Brownian bridge 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6 0 0.2 0.4 0.6 0.8 1 H 1 = max[x(τ), 0 τ 1] τ π H 1 = 8 Brownian excursion 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0-0.2 0 0.2 0.4 0.6 0.8 1 H 1 = max[x(τ), 0 τ 1] τ π H 1 = 2 de Bruijn, Knuth, Rice 72 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 8 / 44

Numerical computation of H N N. Bonichon, M. Mosbah (2003) H N = max τ [x N (τ), 0 τ 1] H N num 0.82 N H N num 1.67 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 9 / 44

Main results 1 Connection between watermelons and random matrices = exact asymptotic results for H N for N 1 H N N H N 2 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 10 / 44

Main results 2 Exact results for the full distribution of H N Cumulative distribution F N (M) F 1 ( 2 11/6 N 1/6 M 2N ) F N (M) = Proba[H N M], N F 1 is the Tracy-Widom distribution for GOE random matrices G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 11 / 44

Outline 1 Vicious walkers and random matrices 2 Connection with stochastic growth models 3 Exact computation using Feynman-Kac formula 4 Large N limit using discrete orthogonal polynomials 5 Conclusion G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 12 / 44

Non intersecting Brownian motions and RMT x i (t) 0 τ 1 t Joint probability of x 1 (τ), x 2 (τ),, x N (τ) at fixed time τ P joint (x 1, x 2,, x N, τ) σ(τ) = 2τ(1 τ) N i<j=1 (x i x j ) 2 e 1 N σ 2 (τ) i=1 x 2 i The rescaled positions x i σ(τ) are distributed like the eigenvalues of random matrices of the Gaussian Unitary Ensemble (GUE, β = 2) G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 14 / 44

Non intersecting Brownian motions and RMT x i(t) 0 τ 1 t H H(t), N N Hermitian matrices from GUE: 1 2 (B mn (t) + i B ) mn (t), m < n, H mn (t) = B mm (t), m = n 1 2 (B nm (t) i B ) nm (t), m > n, where B m,n, B m,n independent Brownian bridges Eigenvalues of H(t) {λ 1 (t) < λ 2 (t) < < λ N (t)} d = {x 1 (t) < x 2 (t) < < x N (t)} G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 15 / 44

Non intersecting Brownian motions and RMT x i (t) 0 τ 1 t Joint probability of x 1 (τ), x 2 (τ),, x N (τ) at fixed time τ P joint (x, τ) N i=1 x 2 i 1 i<j N (xi 2 xj 2 ) 2 e x 2 σ 2 (τ) The rescaled positions x i σ(τ) are distributed like the eigenvalues of random matrices of the Bogoliubov-de Gennes type (class C) A. Atland, M. R. Zirnbauer 96 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 16 / 44

Non intersecting Brownian motions and RMT x i(t) 0 τ 1 t C(t), 2N 2N Herm. rand. mat. of Bologoliubov-de-Gennes type ( ) H(t) S(t) C(t) = S (t) t A. Atland, M. R. Zirnbauer 96 H(t) H(t) is N N Hermitian, S(t) is N N complex symmetric, where the entries are Brownian bridges C(t) is symplectic, C(t) sp(2n), i.e.: ( t Op I C(t) J + J C(t) = 0, J = p I p O p ) G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 17 / 44

Non intersecting Brownian motions and RMT x i(t) 0 τ 1 t C(t), 2N 2N Herm. rand. mat. of Bologoliubov-de-Gennes type ( ) H(t) S(t) C(t) = S (t) t A. Atland, M. R. Zirnbauer 96 H(t) H(t) is N N Hermitian, S(t) is N N complex symmetric, where the entries are Brownian bridges Positive eigenvalues of C(t) d = positions of walkers M. Katori et al. 03 λ N > > λ 2 > λ 1 > λ 1 > λ 2 > λ N, λ i > 0 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 17 / 44

Non intersecting Brownian motions and RMT The rescaled positions x i σ(τ) are distributed like the eigenvalues of random matrices of Gaussian Unitary Ensemble (GUE, β = 2) Mean density ρ(λ) of eigenvalues λ 1, λ 2,, λ N for GUE ρ(λ) = 1 N N δ(λ λ α ) α=1 WIGNER SEMI CIRCLE ρ (λ) SEA TRACY WIDOM N 1/6 (2N) 1/2 0 λ (2N) 1/2 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 19 / 44

Non intersecting Brownian motions and RMT The rescaled positions x i σ(τ) are distributed like the eigenvalues of random matrices of Gaussian Unitary Ensemble (GUE, β = 2) Largest eigenvalue of random matrices from GUE λ max = max 1 i N λ i WIGNER SEMI CIRCLE ρ (λ) TRACY WIDOM = 2 N + N 1 6 2 χ 2 SEA N 1/6 Pr[χ 2 ξ] = F 2 (ξ) Tracy Widom distribution (2N) 1/2 0 λ (2N) 1/2 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 19 / 44

Non intersecting Brownian motions and RMT The rescaled positions x i σ(τ) are distributed like the eigenvalues of random matrices of Gaussian Unitary Ensemble (GUE, β = 2) Largest eigenvalue of random matrices from GUE λ max = max 1 i N λ i Pr[χ 2 ξ] = F 2 (ξ) = 2 N + N 1 6 2 χ 2 Tracy Widom distribution [ F 2 (ξ) = exp ξ ] (s ξ)q 2 (s) ds where q(s) satisfies Painlevé II q (s) = s q(s) + 2q 3 (s) q(s) Ai(s), s C. Tracy, H. Widom 94 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 19 / 44

Watermelons in the limit of large N Consequences for watermelons without wall for large N x N (τ) 2τ(1 τ) 2N + N 1/6 2 χ 2 Proba[χ 2 ξ] = F 2 (ξ), Tracy-Widom distribution for β = 2 When N, x N (τ) reaches a deterministic elliptic shape x N (τ) 2 N τ(1 τ) 0 τ 1 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 20 / 44

Asymptotic behavior of H N : without wall Consequences for watermelons without wall for large N x N (τ) 2τ(1 τ) 2N + N 1/6 2 χ 2 x N (τ) 2 N τ(1 τ) 0 τ 1 The maximal height is reached for τ = 1/2 H N = max[x N (τ), 0 τ 1] τ H N = x N (τ = 1 2 ) N vs. H N num 0.82 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 21 / 44

Asymptotic behavior of H N : with wall Consequences for watermelons without wall for large N x N (τ) 2τ(1 τ) 2 N + N 1/6 2 2/3 χ 2 Proba[χ 2 ξ] = F 2 (ξ), Tracy-Widom distribution for β = 2 The maximal height is reached for τ = 1/2 H N = max[x N (τ), 0 τ 1] τ H N = x N (τ = 1 2 ) 2N vs. H N num 1.67 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 22 / 44

Summary I Connection between watermelons and random matrices = exact asymptotic results for H N for N 1 H N N H N 2 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 23 / 44

Summary I Connection between watermelons and random matrices = exact asymptotic results for H N for N 1 H N N H N 2 N What about the fluctuations of H N? G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 23 / 44

Fluctuations of H N in the limit of large N Consequences for watermelons without wall for large N x N (τ) 2τ(1 τ) 2N + N 1/6 2 χ 2 Proba[χ 2 ξ] = F 2 (ξ), Tracy-Widom distribution for β = 2 When N, x N (τ) reaches a deterministic elliptic shape x N (τ) 2 N τ(1 τ) Fluctuations 0 τ 1 x N (τ = 1/2) N N 1 6 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 24 / 44

Curved growing interface : the PNG droplet Polynuclear Growth Model t = 0 t = t 1 > 0 t = t 2 > t 1 seed G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 26 / 44

Curved growing interface : the PNG droplet Polynuclear Growth Model t = 0 t = t 1 > 0 t = t 2 > t 1 seed At large time t the profile becomes droplet-like h(x, t) h(x, t) 2t 1 (x/t) 2 t +t x G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 26 / 44

Curved growing interface : the PNG droplet Polynuclear Growth Model t = 0 t = t 1 > 0 t = t 2 > t 1 seed At large time t the profile becomes droplet-like h(x, t) h(x, t) 2t 1 (x/t) 2 Fluctuations : KPZ equation t h(x, t) = ν 2 h(x, t) + λ 2 ( h(x, t))2 + ζ(x, t) ζ(x, t)ζ(x, t ) = Dδ(x x )δ(t t ) t +t x G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 26 / 44

Curved growing interface : the PNG droplet Fluctuations : focus on extreme statistics M h(x, t) t X M +t x G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 27 / 44

Curved growing interface : the PNG droplet Fluctuations : focus on extreme statistics M h(x, t) KPZ scaling M 2t t 1/3 X M t 2/3 t X M +t x G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 27 / 44

Curved growing interface : the PNG droplet Fluctuations : focus on extreme statistics M h(x, t) KPZ scaling M 2t t 1/3 X M t 2/3 t X M +t x What is the joint distribution of M, X M? G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 27 / 44

Connection with the Directed Polymer (DPRM) DP in random media with one free end ( point to line ) E(C) = i,j C ɛ ij t X M x M Energy of the optimal polymer X M Transverse coordinate of the optimal polymer G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 28 / 44

Vicious walkers and PNG droplet watermelons PNG droplet x N (τ) 2 N τ(1 τ) h(x, t) h(x, t) 2t 1 (x/t) 2 0 τ 1 t +t x x N h τ x N t G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 29 / 44

Vicious walkers and PNG droplet watermelons PNG droplet x N (τ) 2 N τ(1 τ) h(x, t) h(x, t) 2t 1 (x/t) 2 0 τ 1 t +t x h(ut 2 3, t) 2t t 1 3 d = [ 2 x N ( 1 2 + u 2 N 1 3 ) ] N N 1 6 d = A 2 (u) u 2 Prähofer & Spohn 00 A 2 (u) Airy 2 process G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 29 / 44

Vicious walkers and PNG droplet Use this correspondence to study extreme statistics of PNG M x i (τ) N = 4 M h(x, t) 0 τ M τ 1 HERE: exact computation of the distribution P N (M) t X M +t x P N (M) in the N limit G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 30 / 44

Transition probability x i (t) u 4 v 4 P N (v 1, v 2,, v N, t 2 u 1, u 2,, u N, t 1 ) =? u 3 u 2 u 1 t 1 t 2 v 3 v 2 v 1 The case N = 2: a reflection principle Karlin-Mc Gregor (1959), Lindström (1973), Gessel-Viennot (1985) t P 2 (v 1, v 2, t 2 u 1, u 2, t 1 ) = P 1 (v 1, t 2 u 1, t 1 )P 1 (v 2, t 2 u 2, t 1 ) P 1 (v 2, t 2 u 1, t 1 )P 1 (v 1, t 2 u 2, t 1 ) = P 2 (v 1, v 2, t 2 u 1, u 2, t 1 ) P 2 (v 2, v 1, t 2 u 1, u 2, t 1 ) where P 2 (..) transition probability for two free particles (allowed to cross each other) G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 32 / 44

Watermelons configurations : regularization procedure Brownian motion has an infinite density of zero-crossings H 4 x i (t) Such configurations are ill-defined for Brownian motion : x i (0) = x i+1 (0) 0 1 t AND x i (t = 0 + ) < x i+1 (t = 0 + ) G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 33 / 44

Watermelons configurations : regularization procedure Brownian motion has an infinite density of zero-crossings H 4 x i (t) Such configurations are ill-defined for Brownian motion : x i (0) = x i+1 (0) 0 1 t AND x i (t = 0 + ) < x i+1 (t = 0 + ) A need for regularization : introduce cut-offs ɛ i s ǫ 4 ǫ 3 ǫ 2 ǫ 4 ǫ 3 ǫ 2 Only at the end take the limit ɛ i 0 ǫ 1 ǫ 1 0 1 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 33 / 44

Distribution of the maximal height : without wall Cumulative distribution of the maximal height M F N (M) = Pr [x N (τ) M, 0 τ 1] ǫ 4 ǫ 3 ǫ 4 ǫ 3 ǫ 2 ǫ 2 ǫ 1 ǫ 1 0 1 Path integral (Feynman-Kac formula) for free fermions 2) F N (M) = 2 (N N 1 j=0 j! det [( 1) i 1 H i+j 2 (0) H i+j 2 ( 2M)e 2M2] 1 i,j N where H n (M) Hermite Polynomials see also T. Feierl 08 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 34 / 44

Distribution of the maximal height : without wall 2) F N (M) = 2 (N N 1 j=0 j! det [( 1) i 1 H i+j 2 (0) H i+j 2 ( 2M)e 2M2] 1 i,j N Shape and asymptotic behavior 1.4 1.2 without wall N=2 without wall N=3 F N (M) M N2 +N, M 0 1 F N (M) e 2M2, M F N (M) 1 0.8 0.6 0.4 0.2 0-0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 M G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 35 / 44

Exact value of the H N H 1 = H 2 = H 3 = H 4 =... π 2 4 π ( 1 + ) 2 4 π 96 π 20736 ( 45 + 36 2 8 ) 6 ( 17091 + 5184 2 1888 ) 6 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 36 / 44

Comparison with numerics by Bonichon & Mosbah Numerical estimate by Bonichon & Mosbah H N 2 num 0.82N <H N > 2 20 16 12 8 4 0 0 5 10 15 20 25 N Exact behavior at large N H N 2 N G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 37 / 44

Distribution of the maximal height : with a wall Cumulative distribution of the maximal height G. S., S. N. Majumdar, A. Comtet, J. Randon-Furling 08 F N (M) = A N = A N M 2N2 +N + N n 1,,n N =0 i=1 π 2N2 +N n 2 i 1 j<k N (n 2 j n 2 k) 2 e π2 2M 2 N i=1 n2 i 2 N2 N/2 N 1 j=0 Γ(2 + j)γ( cf Selberg integral 3 2 + j) see also T. Feierl 08 and 12, N. Kobayashi et al. 08 Shape and asymptotic behavior 1.6 1.4 with wall N=2 with wall N=3 F N (M) α N π 2 M 2N2 +N e 1 F N (M) e 2M2, M 12M 2 N(N+1)(2N+1), M 0 F N (M) 1.2 1 0.8 0.6 0.4 0.2 0-0.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 38 / 44 M

Distribution of the maximal height : with a wall Cumulative distribution of the maximal height G. S., S. N. Majumdar, A. Comtet, J. Randon-Furling 08 F N (M) = A N = A N M 2N2 +N + N n 1,,n N =0 i=1 π 2N2 +N n 2 i 1 j<k N (n 2 j n 2 k) 2 e π2 2M 2 N i=1 n2 i 2 N2 N/2 N 1 j=0 Γ(2 + j)γ( cf Selberg integral 3 2 + j) see also T. Feierl 08 and 12, N. Kobayashi et al. 08 What about the asymptotic behavior of F N (M) for N? G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 38 / 44

Discrete orthogonal polynomials Large N analysis of F N (M) = A N M 2N2 +N + N n 1,,n N =0 i=1 n 2 i 1 j<k N (n 2 j n 2 k )2 e π2 2M 2 N i=1 n2 i Discrete orthogonal polynomials n= p k (n)p k (n)e π2 2M 2 n2 = δ k,k h k, p k (n) = n k + Useful expression for asymptotic anaysis F N (M) = B N M 2N2 +N N k=1 h 2k 1 G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 40 / 44

Large N limit for F N (M) For large N, in the "double-scaling limit" P. J. Forrester, S. N. Majumdar, G. S. 11 i.e. d 2 ( dt 2 log F N 2N(1 + t/(2 7/3 N ))) 2/3 = 1 ( ) q 2 (t) q (t) 2 q (t) = 2q 3 (t) + tq(t), q(t) Ai(t), t F N (M) F 1 ( 2 11/6 p 1/6 M 2N ) F 1 (t) = exp ( 1 ( (s t) q 2 (s) q(s) ) ) ds 2 t Tracy-Widom distribution for GOE G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 41 / 44

Large N limit for F N (M) F N (M) F 1 ( 2 11/6 p 1/6 M 2N ) F 1 (t) = exp ( 1 ( (s t) q 2 (s) q(s) ) ) ds 2 t Tracy-Widom distribution for GOE F N (M) TW right tail left tail 2 N M G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 42 / 44

Conclusion Exact results for H N for large N Exact result for distribution of the maximal height H N using path integral techniques Connections with stochastic growth models for large N Large N limit F N (M) F 1 ( 2 11/6 N 1/6 M 2N ), N see also Liechty 11 for a recent rigorous proof G. Schehr (LPTMS Orsay) Max. height of vicious walkers Marseille, Aléa 2012 44 / 44