Vicious Walkers, Random Matrices and 2-d Yang-Mills theory

Transcription

1 Vicious Walkers, Random Matrices and 2-d Yang-Mills theory Laboratoire de Physique Théorique et Modèles Statistiques,CNRS, Université Paris-Sud, France Collaborators: A. Comtet (LPTMS, Orsay) P. J. Forrester (Dept. of Math., Melbourne) C. Nadal (Oxford University, UK) J. Randon-Furling (Univ. Paris 1, Paris) G. Schehr (LPTMS, Orsay)

2 Vicious Walkers, Random Matrices and 2-d Yang-Mills theory Laboratoire de Physique Théorique et Modèles Statistiques,CNRS, Université Paris-Sud, France References: G. Schehr, S. M., A. Comtet, J. Randon-Furling, Phys. Rev. Lett. 1, 1506 (2008) C. Nadal, S. M., Phys. Rev. E 79, (2009) P. J. Forrester, S. M., G. Schehr, Nucl. Phys. B 844, 500 (21) G. Schehr, S. M., A. Comtet, P. J. Forrester, arxiv: (to appear in J. Stat. Phys. (23))

3 Plan: Part I: Nonintersecting Brownian Motions N nonintersecting Brownian excursions half-watermelons Two questions: (i) joint distribution of the positions at fixed time (ii) distribution of the maximal height Exact formula = large N asymptotics via YM 2 = 3-rd order phase transition Part II: Yang-Mills gauge theory in 2-d Summary and Conclusions

4 I : Nonintersecting Brownian Motions

5 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 1 (t) < x 2 (t) <... < x N (t), t 0 x 4(0) x 3(0) x 2(0) x 1(0) 0 t P.-G. De Gennes, 1968, D. A. Huse and M. E. Fisher, 1984,...

6 Fibrous Polymers

7 Non-intersecting Brownian bridges in 1d N Brownian bridges in one-dimension ẋ i (t) = ζ i (t), ζ i (t)ζ j (t ) = δ i,j δ(t t ), 0 t 1 x i (0) = x i (t = 1) = 0 Non-intersecting condition x x 1 (t) < x 2 (t) <... < x N (t) 0 < t < t watermelon Reunion probability Comm.-Incomm. phase transition Huse & Fisher, Fisher (1984),..., Johansson (2002), Ferrari & Praehofer (2006),...

8 Non-intersecting Brownian excursions in 1d N Brownian excursions in one-dimension ẋ i (t) = ζ i (t), ζ i (t)ζ j (t ) = δ i,j δ(t t ), 0 t 1 x i (0) = x i (t = 1) = 0 x i (t) > 0 for 0 < t < 1 Non-intersecting condition x x 1 (t) < x 2 (t) <... < x N (t) 0 < t < 1 0 wall 1 t half watermelon watermelon "with a wall" Katori & Tanemura (2004), Tracy & Widom (2007),...

9 Vicious Walkers in Physics P. G. de Gennes, Soluble Models for fibrous structures with steric constraints (1968) D. A. Huse and M. E. Fisher, Commensurate melting, domain walls, and dislocations (1984); M. E. Fisher, Walks, Walls, Wetting and Melting (1984) B. Duplantier Statistical Mechanics of Polymer Networks of Any Topology (1989) J. W. Essam, A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions (1995) H. Spohn, M. Praehofer, P. L. Ferrari et al. Stochastic growth models (2006) T. L. Einstein et. al., Fluctuating step edges on vicinal surfaces (2004 )... Connection between Vicious Walkers and Random Matrix Theory

10 Fluctuating step edges: Maryland group

11 Brownian excursions and Dyck paths HALF WATERMELON x 0 WALL 1 t In presence of a hard wall at the origin half-watermelons Continuous space-time: Non-intersecting Brownian Excursions Discrete space-time: Dyck paths (combinatorial objects) (Cardy, Katori, Tanemura, Krattenthaler, Fulmek, Feierl, Guttmann, Viennot, Tracy-Widom...)

12 Two questions τ 0 1 time H N maximal height of half watermelon = max [x 0<τ (τ) N ] < 1 Q1: At fixed 0 < τ < 1, what is the joint distribution of positions P joint ({x i } τ)? Q2: What is the probability distribution of the global maximal height Prob.[H N M, N] = F N (M)?

13 Two questions τ 0 1 time H N maximal height of half watermelon = max [x 0<τ (τ) N ] < 1 Q1: At fixed 0 < τ < 1, what is the joint distribution of positions P joint ({x i } τ)? Q2: What is the probability distribution of the global maximal height Prob.[H N M, N] = F N (M)?

14 Method: Path Integral for free fermions space y x Propagator from y at τ = 0 to x at τ = t 0 time t [ x G( x, y, t) = y D x(τ) exp 1 2 t 0 = x e Ĥt y ; where Ĥ 1 2 = E ψ E ( x)ψ E ( y) e E t i ẋ 2 i (τ)dτ ] i 2 x i 1 x1 (τ)<x 2 (τ)< <x N (τ) ψ E ( x) det [φ ni (x j )] Slater determinant (N N) Alternative methods: Lindstrom-Gessel-Viennot method (discrete lattice paths) Karlin-Mcgregor formula (continuous paths)

15 Method: Path Integral for free fermions space y x Propagator from y at τ = 0 to x at τ = t 0 time t [ x G( x, y, t) = y D x(τ) exp 1 2 t 0 = x e Ĥt y ; where Ĥ 1 2 = E ψ E ( x)ψ E ( y) e E t i ẋ 2 i (τ)dτ ] i 2 x i 1 x1 (τ)<x 2 (τ)< <x N (τ) ψ E ( x) det [φ ni (x j )] Slater determinant (N N) Alternative methods: Lindstrom-Gessel-Viennot method (discrete lattice paths) Karlin-Mcgregor formula (continuous paths)

16 Method: Path Integral for free fermions space y x Propagator from y at τ = 0 to x at τ = t 0 time t [ x G( x, y, t) = y D x(τ) exp 1 2 t 0 = x e Ĥt y ; where Ĥ 1 2 = E ψ E ( x)ψ E ( y) e E t i ẋ 2 i (τ)dτ ] i 2 x i 1 x1 (τ)<x 2 (τ)< <x N (τ) ψ E ( x) det [φ ni (x j )] Slater determinant (N N) Alternative methods: Lindstrom-Gessel-Viennot method (discrete lattice paths) Karlin-Mcgregor formula (continuous paths)

17 Method: Path Integral for free fermions space y x Propagator from y at τ = 0 to x at τ = t 0 time t [ x G( x, y, t) = y D x(τ) exp 1 2 t 0 = x e Ĥt y ; where Ĥ 1 2 = E ψ E ( x)ψ E ( y) e E t i ẋ 2 i (τ)dτ ] i 2 x i 1 x1 (τ)<x 2 (τ)< <x N (τ) ψ E ( x) det [φ ni (x j )] Slater determinant (N N) Alternative methods: Lindstrom-Gessel-Viennot method (discrete lattice paths) Karlin-Mcgregor formula (continuous paths)

18 Method: Path Integral for free fermions space y x Propagator from y at τ = 0 to x at τ = t 0 time t [ x G( x, y, t) = y D x(τ) exp 1 2 t 0 = x e Ĥt y ; where Ĥ 1 2 = E ψ E ( x)ψ E ( y) e E t i ẋ 2 i (τ)dτ ] i 2 x i 1 x1 (τ)<x 2 (τ)< <x N (τ) ψ E ( x) det [φ ni (x j )] Slater determinant (N N) Alternative methods: Lindstrom-Gessel-Viennot method (discrete lattice paths) Karlin-Mcgregor formula (continuous paths)

19 Q1: Joint distribution Wishart eigenvalues x At fixed time 0 < τ < 1, let {x 1, x 2,..., x N } positions of walkers 0 τ 1 t P joint ({x i } τ) N i=1 x 2 i j<k (x 2 j [ xk 2 ) 2 1 exp 2τ(1 τ) (Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)) i x 2 i ] x 2 i =λ i eigenvalues of the Wishart matrix W W = X X where X Gaussian random matrix (GUE)

20 Q1: Joint distribution Wishart eigenvalues x At fixed time 0 < τ < 1, let {x 1, x 2,..., x N } positions of walkers 0 τ 1 t P joint ({x i } τ) N i=1 x 2 i j<k (x 2 j [ xk 2 ) 2 1 exp 2τ(1 τ) (Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)) i x 2 i ] x 2 i =λ i eigenvalues of the Wishart matrix W W = X X where X Gaussian random matrix (GUE)

21 Q1: Joint distribution Wishart eigenvalues x At fixed time 0 < τ < 1, let {x 1, x 2,..., x N } positions of walkers 0 τ 1 t P joint ({x i } τ) N i=1 x 2 i j<k (x 2 j [ xk 2 ) 2 1 exp 2τ(1 τ) (Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)) i x 2 i ] x 2 i =λ i eigenvalues of the Wishart matrix W W = X X where X Gaussian random matrix (GUE)

22 Top curve at fixed time: Tracy-Widom (GUE) x topmost curve at fixed time τ: x 2 N (τ) largest eigenvalue of Wishart matrices 0 τ 1 t largest eigenvalue of the Wishart GUE matrix (properly scaled for large N) is distributed via the Tracy-Widom GUE law (Johansson 2000, Johnstone, 20) This shows that the top position x N (τ) typically fluctuates for large N as x N (τ) 2τ(1 τ) = 2 N + 2 2/3 N 1/6 χ 2 where Pr[χ 2 ξ] = F 2 (ξ) Tracy-Widom (GUE)

23 Top curve at fixed time: Tracy-Widom (GUE) x topmost curve at fixed time τ: x 2 N (τ) largest eigenvalue of Wishart matrices 0 τ 1 t largest eigenvalue of the Wishart GUE matrix (properly scaled for large N) is distributed via the Tracy-Widom GUE law (Johansson 2000, Johnstone, 20) This shows that the top position x N (τ) typically fluctuates for large N as x N (τ) 2τ(1 τ) = 2 N + 2 2/3 N 1/6 χ 2 where Pr[χ 2 ξ] = F 2 (ξ) Tracy-Widom (GUE)

24 Tracy-Widom distributions (1994) The scaling function F β (x) has the expression: β = 1: F 1 (x) = exp [ 1 [ 2 x (y x)q 2 (y) + q(y) ] dy ] β = 2: F 2 (x) = exp [ x (y x)q2 (y) dy ] d 2 q dy 2 = 2 q 3 (y) + yq(y) with q(y) Ai(y) as y Painlevé-II Probability density: f β (x) = df β (x)/dx Probability densities f(x) 0.5 β = β = β = x

25 Q2: Maximal height of a watermelon with a wall M x H N.. H N random variable Q: What is its distribution Prob[H N M, N] =F N (M)? 0 1 τ N = 1: F 1 (M) = 2π 5/2 M 3 k=1 k 2 e π2 k 2 /2 M 2 (Chung 75, Kennedy 76) N = 2, F 2 (M) complicated (Katori et. al., 2008)

26 Q2: Maximal height of a watermelon with a wall M x H N.. H N random variable Q: What is its distribution Prob[H N M, N] =F N (M)? 0 1 τ N = 1: F 1 (M) = 2π 5/2 M 3 k=1 k 2 e π2 k 2 /2 M 2 (Chung 75, Kennedy 76) N = 2, F 2 (M) complicated (Katori et. al., 2008)

27 Q2: Maximal height of a watermelon with a wall M x H N.. H N random variable Q: What is its distribution Prob[H N M, N] =F N (M)? 0 1 τ N = 1: F 1 (M) = 2π 5/2 M 3 k=1 k 2 e π2 k 2 /2 M 2 (Chung 75, Kennedy 76) N = 2, F 2 (M) complicated (Katori et. al., 2008)

28 Exact result for all N via Fermionic path integral ǫ 4 ǫ 3 M ǫ 4 ǫ 3 F N (M) = Proba[x N (τ) M, τ [0, 1]] F N (M) = R M(1) R (1) ǫ 2 ǫ 2 ǫ 1 ǫ R M (1) = ɛ e Ĥ ɛ = = E ψ E ( ɛ) 2 e E R M (1) proba. that N walkers return to their initial positions at τ = 1 Ĥ i [ x i + V (x i ) ] potential V (x) = 0 for 0 < x < M = for x = 0, M (Absorbing b.c.) ψ E ( ɛ) det [sin (n i πɛ j /M)] Slater determinant (N N) ( Energy E = π2 2M n n ) n2 N

29 Exact result for all N via Fermionic path integral ǫ 4 ǫ 3 M ǫ 4 ǫ 3 F N (M) = Proba[x N (τ) M, τ [0, 1]] F N (M) = R M(1) R (1) ǫ 2 ǫ 2 ǫ 1 ǫ R M (1) = ɛ e Ĥ ɛ = = E ψ E ( ɛ) 2 e E R M (1) proba. that N walkers return to their initial positions at τ = 1 Ĥ i [ x i + V (x i ) ] potential V (x) = 0 for 0 < x < M = for x = 0, M (Absorbing b.c.) ψ E ( ɛ) det [sin (n i πɛ j /M)] Slater determinant (N N) ( Energy E = π2 2M n n ) n2 N

30 Exact result for all N via Fermionic path integral Using Fermionic path-integral techniques we derived the full Prob. Dist. of H N for all N exactly Cumul. distr: F N (M) = Prob[H N M] [ N F N (M) = (nj 2 nk) 2 2 exp B N M 2N2 +N n i =1,2... i=1 n 2 i j<k π 2N2 +N 2 N/2 N2 where B N = N 1 j=0 Γ(j + 2)Γ(j + 3/2) π2 2M 2 [Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)] Q: Asymptotics of F N (M) as a function of M for large but fixed N? Yang-Mills gauge theory on a 2-d sphere Exploiting this correspondence to YM 2 we find: (Schehr, S.M., Comtet, Forrester, 21/22) i n 2 i ]

31 Exact result for all N via Fermionic path integral Using Fermionic path-integral techniques we derived the full Prob. Dist. of H N for all N exactly Cumul. distr: F N (M) = Prob[H N M] [ N F N (M) = (nj 2 nk) 2 2 exp B N M 2N2 +N n i =1,2... i=1 n 2 i j<k π 2N2 +N 2 N/2 N2 where B N = N 1 j=0 Γ(j + 2)Γ(j + 3/2) π2 2M 2 [Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)] Q: Asymptotics of F N (M) as a function of M for large but fixed N? Yang-Mills gauge theory on a 2-d sphere Exploiting this correspondence to YM 2 we find: (Schehr, S.M., Comtet, Forrester, 21/22) i n 2 i ]

32 Exact result for all N via Fermionic path integral Using Fermionic path-integral techniques we derived the full Prob. Dist. of H N for all N exactly Cumul. distr: F N (M) = Prob[H N M] [ N F N (M) = (nj 2 nk) 2 2 exp B N M 2N2 +N n i =1,2... i=1 n 2 i j<k π 2N2 +N 2 N/2 N2 where B N = N 1 j=0 Γ(j + 2)Γ(j + 3/2) π2 2M 2 [Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)] Q: Asymptotics of F N (M) as a function of M for large but fixed N? Yang-Mills gauge theory on a 2-d sphere Exploiting this correspondence to YM 2 we find: (Schehr, S.M., Comtet, Forrester, 21/22) i n 2 i ]

33 Exact result for all N via Fermionic path integral Using Fermionic path-integral techniques we derived the full Prob. Dist. of H N for all N exactly Cumul. distr: F N (M) = Prob[H N M] [ N F N (M) = (nj 2 nk) 2 2 exp B N M 2N2 +N n i =1,2... i=1 n 2 i j<k π 2N2 +N 2 N/2 N2 where B N = N 1 j=0 Γ(j + 2)Γ(j + 3/2) π2 2M 2 [Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)] Q: Asymptotics of F N (M) as a function of M for large but fixed N? Yang-Mills gauge theory on a 2-d sphere Exploiting this correspondence to YM 2 we find: (Schehr, S.M., Comtet, Forrester, 21/22) i n 2 i ]

34 Exact result for all N via Fermionic path integral Using Fermionic path-integral techniques we derived the full Prob. Dist. of H N for all N exactly Cumul. distr: F N (M) = Prob[H N M] [ N F N (M) = (nj 2 nk) 2 2 exp B N M 2N2 +N n i =1,2... i=1 n 2 i j<k π 2N2 +N 2 N/2 N2 where B N = N 1 j=0 Γ(j + 2)Γ(j + 3/2) π2 2M 2 [Schehr, S.M., Comtet, Randon-Furling, PRL, 1, 1506 (2008)] Q: Asymptotics of F N (M) as a function of M for large but fixed N? Yang-Mills gauge theory on a 2-d sphere Exploiting this correspondence to YM 2 we find: (Schehr, S.M., Comtet, Forrester, 21/22) i n 2 i ]

35 Asymptotic large N results: M F N (M) TW x H N.. right tail left tail 0 1 τ Cum. distr. F N (M) = Prob.[H N M] behaves, for large N as: [ ( )] M exp N 2 Φ 2N for 0 2N 2N M O( N) F 1 [2 11/6 N 1/6 (M ] 2N) for M 2N O(N 1/6 ) [ ( )] M 1 B exp βn Φ + 2N for M 2N O( N) M

36 Asymptotic large N results: where F 1 (x) Tracy-Widom GOE φ ± (x) left and right rate functions = explicitly computable Right rate function: φ + (x) = 4 x x ln Left rate function: (Schehr, S.M., Comtet, Forrester, 21/22) [ ( x ) ] 2x x 1 φ (x) can be expressed in terms of elliptic functions In particular, φ + (x) 29/2 3 (x 1)3/2 as x 1 + φ (x) 16 3 (1 x)3 as x 1 1

37 Asymptotic large N results: where F 1 (x) Tracy-Widom GOE φ ± (x) left and right rate functions = explicitly computable Right rate function: φ + (x) = 4 x x ln Left rate function: (Schehr, S.M., Comtet, Forrester, 21/22) [ ( x ) ] 2x x 1 φ (x) can be expressed in terms of elliptic functions In particular, φ + (x) 29/2 3 (x 1)3/2 as x 1 + φ (x) 16 3 (1 x)3 as x 1 1

38 Asymptotic large N results: where F 1 (x) Tracy-Widom GOE φ ± (x) left and right rate functions = explicitly computable Right rate function: φ + (x) = 4 x x ln Left rate function: (Schehr, S.M., Comtet, Forrester, 21/22) [ ( x ) ] 2x x 1 φ (x) can be expressed in terms of elliptic functions In particular, φ + (x) 29/2 3 (x 1)3/2 as x 1 + φ (x) 16 3 (1 x)3 as x 1 1

39 Asymptotic large N results: where F 1 (x) Tracy-Widom GOE φ ± (x) left and right rate functions = explicitly computable Right rate function: φ + (x) = 4 x x ln Left rate function: (Schehr, S.M., Comtet, Forrester, 21/22) [ ( x ) ] 2x x 1 φ (x) can be expressed in terms of elliptic functions In particular, φ + (x) 29/2 3 (x 1)3/2 as x 1 + φ (x) 16 3 (1 x)3 as x 1 1

40 3-rd order phase transition x=1 x= M / 2N lim 1 ( N N 2 ln F N M = { ) 2N x = φ (x), x < 1 0, x > 1. Since, φ (x) (1 x) 3 3-rd order phase transition = similar to the Douglas-Kazakov transition in large-n 2-d gauge theory

41 3-rd order phase transition x=1 x= M / 2N lim 1 ( N N 2 ln F N M = { ) 2N x = φ (x), x < 1 0, x > 1. Since, φ (x) (1 x) 3 3-rd order phase transition = similar to the Douglas-Kazakov transition in large-n 2-d gauge theory

42 3-rd order phase transition x=1 x= M / 2N lim 1 ( N N 2 ln F N M = { ) 2N x = φ (x), x < 1 0, x > 1. Since, φ (x) (1 x) 3 3-rd order phase transition = similar to the Douglas-Kazakov transition in large-n 2-d gauge theory

43 II : Yang-Mills gauge theory in 2-d

44 Partition function of Yang-Mills theory in 2d Consider a 2-d manifold M. At each point x: a pair of N N matrix A µ (x) (µ = 1, 2) gauge field Partition function: Z M = [DA µ ]e 1 4λ 2 Tr[F µν F µν]d 2 x F µν = µ A ν ν A µ + i[a µ, A ν ] field strength λ coupling strength Under a local gauge transformation: A µ S 1 (x)a µ S(x) i S 1 (x) µ S(x) where S(x) N N matrix that depends on the underlying gauge group G Field strengths transform as F µν S 1 (x)f µν (x)s(x) that keeps the action gauge invariant. Ex: G U(1) : electrodynamics G SU(2) : electro-weak interact o G SU(3) : chromodynamics

45 Partition function of Yang-Mills theory in 2d Consider a 2-d manifold M. At each point x: a pair of N N matrix A µ (x) (µ = 1, 2) gauge field Partition function: Z M = [DA µ ]e 1 4λ 2 Tr[F µν F µν]d 2 x F µν = µ A ν ν A µ + i[a µ, A ν ] field strength λ coupling strength Under a local gauge transformation: A µ S 1 (x)a µ S(x) i S 1 (x) µ S(x) where S(x) N N matrix that depends on the underlying gauge group G Field strengths transform as F µν S 1 (x)f µν (x)s(x) that keeps the action gauge invariant. Ex: G U(1) : electrodynamics G SU(2) : electro-weak interact o G SU(3) : chromodynamics

46 Partition function of Yang-Mills theory in 2d Consider a 2-d manifold M. At each point x: a pair of N N matrix A µ (x) (µ = 1, 2) gauge field Partition function: Z M = [DA µ ]e 1 4λ 2 Tr[F µν F µν]d 2 x F µν = µ A ν ν A µ + i[a µ, A ν ] field strength λ coupling strength Under a local gauge transformation: A µ S 1 (x)a µ S(x) i S 1 (x) µ S(x) where S(x) N N matrix that depends on the underlying gauge group G Field strengths transform as F µν S 1 (x)f µν (x)s(x) that keeps the action gauge invariant. Ex: G U(1) : electrodynamics G SU(2) : electro-weak interact o G SU(3) : chromodynamics

47 Partition function of Yang-Mills theory in 2d Consider a 2-d manifold M. At each point x: a pair of N N matrix A µ (x) (µ = 1, 2) gauge field Partition function: Z M = [DA µ ]e 1 4λ 2 Tr[F µν F µν]d 2 x F µν = µ A ν ν A µ + i[a µ, A ν ] field strength λ coupling strength Under a local gauge transformation: A µ S 1 (x)a µ S(x) i S 1 (x) µ S(x) where S(x) N N matrix that depends on the underlying gauge group G Field strengths transform as F µν S 1 (x)f µν (x)s(x) that keeps the action gauge invariant. Ex: G U(1) : electrodynamics G SU(2) : electro-weak interact o G SU(3) : chromodynamics

48 Partition function of Yang-Mills theory in 2d Consider a 2-d manifold M. At each point x: a pair of N N matrix A µ (x) (µ = 1, 2) gauge field Partition function: Z M = [DA µ ]e 1 4λ 2 Tr[F µν F µν]d 2 x F µν = µ A ν ν A µ + i[a µ, A ν ] field strength λ coupling strength Under a local gauge transformation: A µ S 1 (x)a µ S(x) i S 1 (x) µ S(x) where S(x) N N matrix that depends on the underlying gauge group G Field strengths transform as F µν S 1 (x)f µν (x)s(x) that keeps the action gauge invariant. Ex: G U(1) : electrodynamics G SU(2) : electro-weak interact o G SU(3) : chromodynamics

49 Lattice Regularization: Consider, for instance, the U(N) gauge theory Regularization on the lattice: Z M = du L Z P [U P ] U P = L L plaquette plaquettes U L U 1 U 5 P 2 U 3 P 1 U 4 U2 Z P plaquette partition function (Wilson, 74, Migdal, 75)

50 Heat-kernel action Z M = du L Z P [U P ] U P = L L plaquette plaquettes U L U 1 U 5 P 2 U 3 P 1 U 4 A common choice : Wilson s action ] Z P (U P ) = exp [b N Tr(U P + U P ) U2 Wilson 74 Exact solution of the Partition Function: (Gross & Witten, Wadia, 80)

51 Heat-kernel action Z M = du L Z P [U P ] U P = L L plaquette plaquettes U L U 1 U 5 P 2 U 3 P 1 U 4 A common choice : Wilson s action ] Z P (U P ) = exp [b N Tr(U P + U P ) U2 Wilson 74 Exact solution of the Partition Function: (Gross & Witten, Wadia, 80) Alternative choice : invariance under decimation Migdal s recursion relation du 3 Z P1 (U 1 U 2 U 3 )Z P2 (U 4 U 5U 3 ) = Z P 1 +P 2 (U 1 U 2 U 4 U 5) Z P (U P ) = [ d R χ R (U P ) exp A ] P 2N C 2(R) Migdal 75, Rusakov 90 R

52 Partition function of Yang-Mills theory on the 2d-sphere Partition funct o on M, of genus g, computed with the heat-kernel action Z M = [ d 2 2g R exp A ] 2N C 2(R) R

53 Partition function of Yang-Mills theory on the 2d-sphere Partition funct o on the sphere computed with the heat-kernel action Z M = [ dr 2 exp A ] 2N C 2(R) R

54 Partition function of Yang-Mills theory on the 2d-sphere Partition funct o on the sphere computed with the heat-kernel action Z M = [ dr 2 exp A ] 2N C 2(R) R Irreducible representations R of G are labelled by the lengths of the Young diagrams: If G = U(N) Z M = c N e A N If G = Sp(2N) Z M = ĉ N e A (N+ 1 2 ) N+1 12 n 1,...,n N =0 i<j n 1,...,n N =0 (n i n j ) 2 e 2N A N j=1 n 2 j i<j Nj=1 n 2 j (n 2 i n 2 j ) 2 e A 4N Nj=1 n 2 j

55 Correspondence between YM 2 on the sphere and watermelons Partition function of YM 2 on the sphere with gauge group Sp(2N) Z M = Z(A; Sp(2N)) Z(A; Sp(2N)) = ĉ N e A (N+ 2 1 ) N+1 12 n 1,...,n N =0 N j=1 n 2 j i<j (n 2 i n 2 j ) 2 e A 4N Nj=1 n 2 j Cumulative distribution of the maximal height of watermelons with a wall F N (M) = A + N N n 2 M 2N2 +N j (ni 2 nj 2 ) 2 e π2 Nj=1 2M 2 nj 2 n 1,,n N =0 j=1 i<j ( Z A = 2π2 N M ; Sp(2N) 2 ) (Forrester, S. M., Schehr, 11)

56 Correspondence between YM 2 on the sphere and watermelons Partition function of YM 2 on the sphere with gauge group Sp(2N) Z M = Z(A; Sp(2N)) Z(A; Sp(2N)) = ĉ N e A (N+ 2 1 ) N+1 12 n 1,...,n N =0 N j=1 n 2 j i<j (n 2 i n 2 j ) 2 e A 4N Nj=1 n 2 j Cumulative distribution of the maximal height of watermelons with a wall F N (M) = A + N N n 2 M 2N2 +N j (ni 2 nj 2 ) 2 e π2 Nj=1 2M 2 nj 2 n 1,,n N =0 j=1 i<j ( Z A = 2π2 N M ; Sp(2N) 2 ) (Forrester, S. M., Schehr, 11)

57 Large N limit of YM 2 and consequences for F N (M) Weak-strong coupling transition (3-rd order) in YM 2, Douglas-Kazakov 93 F N (M) TW N critical region right tail M 2 = 2π2 N A weak coupling strong coupling left tail 2 N M π 2 A Critical point A = A c = π 2 corresponds (using A = 2π2 N M 2 ): M = M c = 2N A > A c (Strong Coupling) M < M c = 2N (left tail of H N ) A < A c (Weak Coupling) M > M c = 2N (right tail of H N )

58 Large N limit of YM 2 and consequences for F N (M) Weak-strong coupling transition (3-rd order) in YM 2, Douglas-Kazakov 93 F N (M) TW N critical region right tail M 2 = 2π2 N A weak coupling strong coupling left tail 2 N M π 2 A Critical point A = A c = π 2 corresponds (using A = 2π2 N M 2 ): M = M c = 2N A > A c (Strong Coupling) M < M c = 2N (left tail of H N ) A < A c (Weak Coupling) M > M c = 2N (right tail of H N )

59 Large N limit of YM 2 and consequences for F N (M) Weak-strong coupling transition (3-rd order) in YM 2, Douglas-Kazakov 93 F N (M) TW N critical region right tail M 2 = 2π2 N A weak coupling strong coupling left tail 2 N M π 2 A Critical point A = A c = π 2 corresponds (using A = 2π2 N M 2 ): M = M c = 2N A > A c (Strong Coupling) M < M c = 2N (left tail of H N ) A < A c (Weak Coupling) M > M c = 2N (right tail of H N )

60 Large N limit of YM 2 and consequences for F N (M) In the critical regime, "double-scaling limit", the method of orthogonal polynomials (Gross-Matytsin 94, Crescimanno-Naculich-Schnitzer 96) shows d 2 ( dt log F 2 N 2N(1 + t/(2 7/3 N ))) 2/3 = 1 ( ) q 2 (t) q (t) 2 q (t) = 2q 3 (t) + t q(t), q(t) Ai(t), t F N (M) F 1 ( 2 11/6 N 1/6 M 2N ) F 1 (t) = exp ( 1 2 t ( ) ) (s t) q 2 (s) + q(s) ds Tracy-Widom distribution for β = 1

61 Large N limit of YM 2 and consequences for F N (M) In the critical regime, "double-scaling limit", the method of orthogonal polynomials (Gross-Matytsin 94, Crescimanno-Naculich-Schnitzer 96) shows d 2 ( dt log F 2 N 2N(1 + t/(2 7/3 N ))) 2/3 = 1 ( ) q 2 (t) q (t) 2 q (t) = 2q 3 (t) + t q(t), q(t) Ai(t), t F N (M) F 1 ( 2 11/6 N 1/6 M 2N ) F 1 (t) = exp ( 1 2 t ( ) ) (s t) q 2 (s) + q(s) ds Tracy-Widom distribution for β = 1 double scaling regime [A A c] Tracy-Widom [M 2N] Forrester, S. M., Schehr, 11

62 Absorbing boundary condition SP(2N) Ratio of reunion probabilities for N vicious walkers on the segment [0, M] with absorbing boundary conditions ǫ 4 ǫ 3 ǫ 2 ǫ 1 ǫ M ǫ 4 ǫ 3 ǫ 2 F N (M) = Proba[x N (τ) M, τ [0, 1]] F N (M) = R M(1) R (1) R M (1) proba. that N walkers return to their initial positions at τ = 1 Related to YM 2 on the sphere with gauge group Sp(2N) ( ) F N (M) Z A = 2π2 N M ; Sp(2N) 2 limiting form of F N (M): F 1 Tracy-Widom (GOE)

63 Periodic boundary condition U(N) Ratio of reunion probabilities for N vicious walkers on the segment [0, M] with periodic boundary conditions ǫ 4 ǫ 3 ǫ 2 ǫ 1 ǫ M ǫ 4 ǫ 3 ǫ 2 F N (M) = Proba[x N (τ) M, τ [0, 1]] F N (M) = R M(1) R (1) R M (1) proba. that N walkers return to their initial positions at τ = 1 Related to YM 2 on the sphere with gauge group U(N) ( ) F N (M) Z A = 4π2 N M ; U(N) 2 limiting form of F N (M): F 2 Tracy-Widom(GUE)

64 Reflecting boundary condition SO(2N) Ratio of reunion probabilities for N vicious walkers on the segment [0, M] with reflecting boundary conditions ǫ 4 ǫ 3 ǫ 2 ǫ 1 ǫ M ǫ 4 ǫ 3 ǫ 2 F N (M) = Proba[x N (τ) M, τ [0, 1]] F N (M) = R M(1) R (1) R M (1) proba. that N walkers return to their initial positions at τ = 1 Related to YM 2 on the sphere with gauge group SO(2N) ( ) F N (M) Z A = 4π2 N M ; SO(2N) 2 limiting form of F N (M): F 2 F 1

65 Summary τ 0 1 time H N = max [x N 0< τ<1 (τ)] x i (τ) trajectory of the i-th walker x N (τ) trajectory of the top path x N (τ) (centered and scaled) Airy 2 process minus a parabola At fixed time τ, the marginal x N (τ) (centered and scaled) Tracy-Widom β = 2 Prähofer & Spohn, 00 However, the maximal height H N = max 0 τ 1 [x N (τ)] (centered and scaled) Tracy-Widom β = 1 an indirect proof via PNG growth model Johansson (2003) Airy 2 process by Moreno Flores, Quastel, Remenik: vicious walker problem using Riemann-Hilbert: Liechty, (22) beautiful connection to YM 2 and the 3-rd order phase transition

66 Summary τ 0 1 time H N = max [x N 0< τ<1 (τ)] x i (τ) trajectory of the i-th walker x N (τ) trajectory of the top path x N (τ) (centered and scaled) Airy 2 process minus a parabola At fixed time τ, the marginal x N (τ) (centered and scaled) Tracy-Widom β = 2 Prähofer & Spohn, 00 However, the maximal height H N = max 0 τ 1 [x N (τ)] (centered and scaled) Tracy-Widom β = 1 an indirect proof via PNG growth model Johansson (2003) Airy 2 process by Moreno Flores, Quastel, Remenik: vicious walker problem using Riemann-Hilbert: Liechty, (22) beautiful connection to YM 2 and the 3-rd order phase transition

67 Open Questions and related issues: boundary conditions gauge groups deeper understanding needed Other interesting observables: Joint distribution of the maximal height H N = max 0 τ 1 [x N (τ)] and the time τ M at which it occurs: P N (H N = M, τ M ) = Interesting relation to KPZ interfaces and (1 + 1)-d directed polymers Rambeau & Schehr 11, Flores et. al. 12, Schehr 12, Quastel & Remenik, 12, Baik, Liechty, Schehr, 12 distribution of the maximal height H 1 (N) = max 0 τ 1 [x 1 (τ)] of the first (lowest) walker?.

68 Open Questions and related issues: boundary conditions gauge groups deeper understanding needed Other interesting observables: Joint distribution of the maximal height H N = max 0 τ 1 [x N (τ)] and the time τ M at which it occurs: P N (H N = M, τ M ) = Interesting relation to KPZ interfaces and (1 + 1)-d directed polymers Rambeau & Schehr 11, Flores et. al. 12, Schehr 12, Quastel & Remenik, 12, Baik, Liechty, Schehr, 12 distribution of the maximal height H 1 (N) = max 0 τ 1 [x 1 (τ)] of the first (lowest) walker?.

69 Open Questions and related issues: boundary conditions gauge groups deeper understanding needed Other interesting observables: Joint distribution of the maximal height H N = max 0 τ 1 [x N (τ)] and the time τ M at which it occurs: P N (H N = M, τ M ) = Interesting relation to KPZ interfaces and (1 + 1)-d directed polymers Rambeau & Schehr 11, Flores et. al. 12, Schehr 12, Quastel & Remenik, 12, Baik, Liechty, Schehr, 12 distribution of the maximal height H 1 (N) = max 0 τ 1 [x 1 (τ)] of the first (lowest) walker?.

70 Open Questions and related issues: boundary conditions gauge groups deeper understanding needed Other interesting observables: Joint distribution of the maximal height H N = max 0 τ 1 [x N (τ)] and the time τ M at which it occurs: P N (H N = M, τ M ) = Interesting relation to KPZ interfaces and (1 + 1)-d directed polymers Rambeau & Schehr 11, Flores et. al. 12, Schehr 12, Quastel & Remenik, 12, Baik, Liechty, Schehr, 12 distribution of the maximal height H 1 (N) = max 0 τ 1 [x 1 (τ)] of the first (lowest) walker?.

71 Consequences for curved stochastic growth M h(x, t) t XM x +t Distribution of the height field h(0, t) 00) ( h(0, t) 2t lim P t t 1/3 ) s = F 2 (s) F 2 (s) Tracy Widom distribution for β = 2 (Prähofer & Spohn,

72 Consequences for curved stochastic growth M h(x, t) t XM x +t Maximum M max t x t h(x, t) (Forrester, S.M. and Schehr, NPB 11) ( ) M 2t lim P s = F 1 (s) t t 1/3 F 1 (s) Tracy Widom distribution for β = 1

73 Consequences for curved stochastic growth Maximum M max t x t h(x, t) (Forrester, S.M., Schehr, NPB 11) ( ) M 2t lim P s = F 1 (s) t t 1/3 F 1 (s) Tracy Widom distribution for β = 1 see also Krug et al. 92, Johansson 03 (indirect proof), G. M. Flores, J. Quastel, D. Remenik, arxiv:

74 Experiments on nematic liquid crystals K. A. Takeuchi, M. Sano, Phys. Rev. Lett. 104, 2306 (20) Extreme-Value Statistics (circular) radius : GUE-TW max height : GOE-TW distribution!! max radius : Gumbel dist. Max heights of circular interfaces obey the GOE-TW dist.! Courtesy of K. Takeuchi