Partition function of one matrix-models in the multi-cut case and Painlevé equations. Tamara Grava and Christian Klein

Transcription

1 Partition function of one matrix-models in the multi-cut case and Painlevé equations Tamara Grava and Christian Klein SISSA IMS, Singapore, March 2006

2 Main topics Large N limit beyond the one-cut case in Hermitian one-matrix model Modulation equations Double scaling limits and Painlevé equations Connection with the theory of nonlinear waves and numerical simulations

3 where One-Hermitian Matrix Integrals Z N (t) = 1 V ol(u N ) N N V t (M) = 1 2 M 2 + k 3 dme NT rv t(m) t k M k is a τ-function for the Toda lattice. For N and t 1, the partition function has the topological large N expansion (Bessis-Itzykson-Zuber 1980, McLaughlin-Ercolani 2002, Bleher-Its 2004) where F 0 (t) = 1 N 2 log Z N(t) Z N (0) = F 0(t) + F 1(t) N 2 sup { R dµ=1 V t (λ)dµ(λ) + + F 2(t) N } log λ η dµ(λ)dµ(η) and the equilibrium density of eigenvalues ψ(λ)dλ is one interval.

4 where F g (t) = n Topological meaning k 1,...,k n ( ) n a g (k 1,..., k n )t k1,... t kn a g (k 1,..., k n ) = Γ 1 Γ where Γ = a connected fat graph of genus g with n vertices of valencies k 1,..., k n. For example g = 1, one vertex, valency 4

5 where Determination of the equilibrium measure R(λ) = ψ(λ)dλ = R( R(λ)h(λ) 2πi )dλ 2m (λ β k ), h(λ) = Res k=1 ξ= V (ξ) R(ξ)(ξ λ). The number m of intervals of the support is uniquely determined by the requirement that F 0 is the absolute minimum. The dependence of β j = β j (t) s on the times t k is given by hyperbolic PDE s called modulation equations for the Toda lattice. For t 2 = 1, t k>2 1, the support of ψ consists of one interval (β 1 (t), β 2 (t)) (Chen-Ismail). Varying the times t k>2 there is a critical time t c when the solution β 1,2 (t c ) reaches a point of gradient catastrophe (F 0 is not anymore the absolute minimum) after which the support of ψ(λ) becomes two or many intervals.

6 Beyond the one-cut case When supp[ψ(λ)] = many intervals, the partition function has no regular large N topological expansion. Oscillatory terms appear (Bonnet-David-Eynard 2000, Deift-Kriecherbauer-McLaughlin-Venakides-Zhou ). Transition from one cut to two cuts double scaling limit of the partition function. Painlevé equations appear (Periwal-Shevitz, Douglas-Seiberg-Shenker, Moore, Brezin-Marinari-Parisi, Fokas-Its-Kitaev, Bleher-Its,... )

7 When the equilibrium measure ψ(λ)dλ is two-cut regular! 1! 2! 3! 4 Bonnet-David-Eynard 2000 derived by mean-field approach Z N (t) e N 2 F 0 (t) θ 3 (NΩ; τ) + O(1) with Ω = β2 β 1 ψ(λ)dλ modular parameter τ = β2 dλ R(λ) / β3 dλ R(λ), β 1 β 2 and R(λ) = (λ β 1 )(λ β 2 )(λ β 3 )(λ β 4 ).

8 In the physics literature the expansion of the free energy in the multi-cut case has been evaluated to all orders using the loop equations (Eynard, Eynard-Cekhov) Z N (t) e N 2 (F 0 (t)+ 1 N 2 F 1(t)+ 1 θ 3 (NΩ; τ) + R k (N, t) N k k 1 N 4 F 2(t)+... ) the topological meaning of the quantities F k (t) is still not known.,

9 Large N asymptotics of recurrence coefficients of orthogonal polynomials P n (z)p m (z)e NV t(z) dz = δ nm, zp n (z) = γ n+1 P n+1 (z) + β n P n (z) + γ n P n 1 (z), γn 2 (t) = 1 2 N 2 t 2 1 When the supp(ψ(λ)) = [β 1, β 2 ] γ 2 N(t) = 1 N 2 2 t 2 1 log Z N (t) log e N 2 (F 0 (t)+ 1 N 2 F 1(t)+ 1 N 4 F 2(t)+... ) When the supp(ψ(λ)) = [β 1, β 2 ] [β 3, β 4 ] (DKMVZ 1999, BDE 2000) γ 2 N(t) = 1 N 2 2 t 2 1 log(e N 2 F 0 (t) θ 3 (NΩ; τ)) + k 1 r k (N, t) N k (0.1)

10 Remark The formula (0.1) can be generalized to the multi-cut case by verifying that (Chen, G.) γ 2 N = 1 N 2 2 t 2 1 log [ ] e N 2 F 0 θ (NΩ, B) + O(1/N), γ 2 N(β N 1 + β N ) = 1 N 2 2 t 1 t 2 log [ ] e N 2 F 0 θ (NΩ, B) + O(1/N), and so on. However it is not mathematically obvious to integrate the equations with respect to the times and obtain Z N e N 2 F 0 θ (NΩ, B).

11 Non regular cases - double scaling limits (A) ψ(λ) (λ v) 2! 1 v! 2 γ 2 N = (γ0 N )2 + N 1 3 q(z)cos(2πnφ(t)) + O(N 2/3 ) where q(z) is the Hastings McLeod solution of PII q zz = zq + 2q 3 (Bleher-Its, Bleher-Eynard, Kuijlaars-Clays Vanlessen,... ) (B) ψ(λ) (λ v)! 1! 2 v (C) ψ(λ) (λ β 2 ) 5 2! 1! 2 Second equation in the Painlevé I hierarchy, P14 z = 40q (q 2 z + 2qq z z) q zzzz (Brezin, Marinari and Parisi).

12 Numerical example (Jurkiewicz 1991, Klein-G.) V (λ) = 1 2 λ t 4λ λ6 The coefficients β n = 0 and the γ n s are determined recursively from the string equation n N = γ n[1 + (t 4 + γ n )(γ n 1 + γ n + γ n+1 ) + γ n 1 γ n γ n 1 (γ n 2 + γ n 1 + γ n ) + γ n+1 (γ n + γ n+1 + γ n+2 )] and (BDE) γ n 1 16 (β 1 + β 3 β 2 β 4 ) 2 θ3(0) 2 θ3 2(2u ) θ 3 (Nω n+1 )θ 3 (Nω n 1 ) θ 3 (Nω n ) u = β 4 ψ(λ)dλ, ω n = Ω + 2(1 n N )u

13 t 4 = t c 4 = 10 3, ( n N ) c = N=400 N=800

14 Work in Progress The comparison of the numerical evaluation of γ N with γ 2 N = 2 regular t 2 1 log(e N 2 F 0 (t) θ 3 (NΩ; τ)) when ψ(λ) is two cut multiscale expansion in terms of Painlevé II, p zz = zp + 2p 3 when ψ(λ) =! 1 v! 2 P14 z = 40p (p 2 z + 2pp z z) p zzzz when ψ(λ) =! 1! 2???? when ψ(λ) =! 1! 2 v

15 Small dispersion limit of the Korteweg de Vries (KdV) equation { ut + 6uu x + ɛ 2 u xxx = 0, x, t, u R, u(x, t = 0, ɛ) = u 0 (x). These oscillations were called Dispersive Shock Waves by Gurevich and Pitaevskii (1973).

16 Different values of ɛ &!*&!&#$ &!*&!% #$ #$ ) )!#$!#$!&!!!"#$!"!%#$!% (!&!!!"#$!"!%#$!% ( &!*&!%#$ &!*&!" #$ #$ ) )!#$!#$!&!!!"#$!"!%#$!% (!&!!!"#$!"!%#$!% (

17 The limit u(x, t, ɛ) as ɛ 0 is determined from the Lax-Levermore maximization problem by the relations F 0 (x, t) = 1 π inf ψ A {(a(x, t, u 0), ψ) (Lψ, ψ)} Lψ(η) = 1 π log η µ η + µ ψ(µ)dµ 1. when supp ψ = [0, u H (x, t)], t u H 6u H x u H = 0 u(x, t, ɛ) = 2ɛ 2 2 x log 2 e F 0(x,t)/ɛ 2 + O(ɛ 2 ) = u H (x, t) + O(ɛ 2 ), 2. when supp ψ = two intervals [ ] u(x, t, ɛ) 2ɛ 2 2 x 2 log e F 0(x,t)/ɛ 2 θ(ω(x, t)/ɛ; τ). Note that the width of the oscillatory zone does not depend on ɛ.

18 The blue line is KdV and the purple the asymptotic solution

19 Difference of KdV and asymptotic solution (t=0.4) "(&%!&"# "(&%!$ %"& %"& %"%# %"%# % %!!%"%#!!%"%#!%"&!%"&!%"&#!%"&#!%"$!!"#!!!$"#!$!%"$!!"#!!!$"#!$ "(&%!$"# "(&%!! %"& %"& %"%# %"%# % %!!%"%#!!%"%#!%"&!%"&!%"&#!%"&#!%"$!!"#!!!$"#!$!%"$!!"#!!!$"#!$

20 Error In the blue zone the error decreases like ɛ In the green zone on the left the error 3 decreases like ɛ In the green zone on the right the error decreases like ɛ

21 Oscillations go to zero, error O( 3 ɛ) and ψ(λ) (λ v) 2! 1 v! 2 The following ansatz holds in the neighbourhood of the trailing edge x = ν(t), and y = (x ν(t))ɛ 2 3 u(x, t, ɛ) = u H (x, t) x=ν(t) + ɛ 1 3 q(y) cos ( φ(x, t) ɛ ) + O(ɛ 2 3 ) where φ(x, t) = s(t)+r(t)(x ν(t)), 3r 2 (t) = 6u H (ν(t), t) ν t (t), s t = 2r 3 and the function q(y) is the Hastings Mcleod solution of the Painlevé II, q yy + yq [r2 (t) 4u H (ν(t), t)] t 6r 2 (t) = 1 2r 2 (t) q3.

22 Multiscale solution PII and KdV solution

23 Difference between KdV and PII solution 0.1 "=10! "=10! ! 0! 0!0.05!0.05!0.1!3.6!3.4!3.2!3 x!0.1!3.6!3.4!3.2!3 x 0.1 "=10! "=10! ! 0! 0!0.05!0.05!0.1!3.6!3.4!3.2!3 x!0.1!3.6!3.4!3.2!3 x

24 Difference between KdV and asymptotic solutions )" )"(% )"( )")%! )!)")%!)"(!)"(%!)"!!"#!!"$!!"%!!"&!!"!!!"!!"(!! *

25 Combination of asymptotic solutions "% "*(!% "(! %!"(!"% "%!!"#!!"$!!"%!!!%"&!%"#!%"$!%"% ) "*(!! "(! %!"(!"%!!"#!!"$!!"%!!!%"&!%"#!%"$!%"%

26 KdV and asymptotic solution 1,2 at breakup for ϵ= ,("!# +,("!%!("&!("& *!("# *!("#!("%!("%!!!!"#!!"$%!!"$#!!"$&!!"$!!"$ )!!!!"#!!"$%!!"$#!!"$&!!"$!!"$ ) +,("( +,("!("&!("& *!("# *!("#!("%!("%!!!!"#!!"$%!!"$#!!"$&!!"$!!"$ )!!!!"#!!"$%!!"$#!!"$&!!"$!!"$ )

27 Non regular cases - Painlevé 14 equation Breakup time (x c, t c ) for the solution u H (x, t) of the Hopf equation u Ht + 6uu Hx = 0; ψ(λ) (λ u H ) 5 2 u H 0 Conjecture: near the critical point the solution behaves as (Dubrovin) ( u(x, t, ɛ) = ɛ 2 x 7 U ɛ 6 7, ) t ɛ O(ɛ 4 7 ) where U(X, T ), X = x/ɛ 6 7, T = t/ɛ 4 7 solves the equation X = 6T U + U (U 2 X + 2UU XX ) U XXXX (Brezin-Marinari-Parisi 92 and Kudashev-Suleimanov 96).

28 Open problems Prove existence and uniqueness of the solution of X = 6T U + U (U X 2 + 2UU XX ) U XXXX such that U(X) ± 3 X as X ± Prove that the KdV solution u(x, t, ɛ), and the recurrence coefficients in random matrix are well described by P14 in the neighbourhood of the breakup time.

29 Numerical proof: KdV and P14 solution & )*&"+!, & )*&"+!$ & )*&"+!- (!&"% (!&"% (!&"%!!!!"#!!"$%!!"$!!"%%!!"% )*&"++& &!!!!"#!!"$%!!"$!!"%%!!"% )*&"+++ &!!!!"#!!"$%!!"$!!"%%!!"% )*&"++, & (!&"% (!&"% (!&"%!!!!"#!!"$%!!"$!!"%%!!"% )*&"++$ &!!!!"#!!"$%!!"$!!"%%!!"% )*&"++- &!!!!"#!!"$%!!"$!!"%%!!"% )*&"+.& & (!&"% (!&"% (!&"%!!!!"#!!"$%!!"$!!"%%!!"%!!!!"#!!"$%!!"$!!"%%!!"%!!!!"#!!"$%!!"$!!"%%!!"%

30 Conclusions and open problems The behaviour of any Hamiltonian perturbation of the simplest hyperbolic PDE s u t + a(u)u x near the critical point of the unperturbed solution is independent from the choice of the perturbation and from the choice of the initial data and it is governed by the fourth order equation P14, (Dubrovin, math-ph/ ). Matrix models suggest that possibly the same statement is true for two components hyperbolic systems. Does Painlevé II share the same universality properties? Determination of the double scaling limit near the critical point (B). Numerically the rescaling does not follow a power law. Random Matrix theory has inspired the study of the typical behaviour of the perturbation of an hyperbolic PDE near the break-up time. Possible results for elliptic equations?

31 Thank you for your attention!