[Paper Review] Caustics, counting maps and semi-classical asymptotics

Nicholas M. Ercolani|arXiv (Cornell University)|Dec 10, 2009

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper establishes that the genus expansion coefficients of Hermitian random matrix partition functions are rational functions of an algebraic function $ z_0(t) $, which generates generalized Catalan numbers and solves the inviscid Burgers equation. The formation of caustics (shock singularities) in the Burgers solution directly corresponds to poles in these rational generating functions, enabling a precise link to the asymptotic expansion of the first Painlevé transcendent via partial fractions of the rational forms.

ABSTRACT

This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. The coefficients of the large N expansion of the logarithm of this partition function,also known as the genus expansion, (and its derivatives) are generating functions for a variety of graphical enumeration problems. The main results are to prove that these generating functions are in fact specific rational functions of a distinguished irrational (algebraic) function of the generating function parameters. This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of parameter). It is also a solution of the inviscid Burgers equation for certain initial data. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions. These results in turn provide new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights. One gains new insights into the relation between certain derivatives of the genus expansion and the asymptotic expansion of the first Painleve transcendent, related to the double-scaling limit. This work provides a precise expression of the Painleve asymptotic coefficients directly in terms of the coefficients of the partial fractions expansion of the rational form of the generating functions established here. Moreover, these insights point toward a more general program relating the first Painleve hierarchy and the higher order structure of the double-scaling limit to the specific rational structure of generating functions.

Motivation & Objective

• To understand the combinatorial and analytic structure of the genus expansion coefficients in the large-$ N $ asymptotic expansion of Hermitian random matrix partition functions.
• To show that these coefficients are rational functions of a single algebraic function $ z_0(t) $, the generating function for generalized Catalan numbers.
• To connect the shock formation (caustics) in the inviscid Burgers equation to the poles of the rational generating functions.
• To establish a precise link between the double-scaling limit of the genus expansion and the asymptotic coefficients of the first Painlevé transcendent.
• To extend the uniform validity of asymptotic expansions for recurrence coefficients and correlation functions to a semi-infinite strip in the complex $ t $-plane.

Proposed method

• Derive the genus expansion of the logarithm of the partition function $ Z_N(t) $ as a series in $ N^{-2} $, with coefficients $ e_g(t) $.
• Identify $ z_0(t) $ as the generating function for generalized Catalan numbers and show it satisfies the inviscid Burgers equation with specific initial data.
• Prove that for $ g \geq 2 $, the coefficients $ e_g(t) $ lie in the function field $ \mathbb{Q}(z_0(-t)) $, i.e., are rational functions of $ z_0(-t) $.
• Use partial fractions decomposition of these rational functions to derive recursive constructions for the coefficients of the generating functions.
• Apply Riemann-Hilbert problem techniques to analyze the asymptotics of orthogonal polynomials and correlation functions, particularly in the double-scaling limit.
• Establish uniform asymptotic expansions for $ \log Z_N(t) $ and recurrence coefficients $ b_{N,N}^2(t) $ in a complex half-strip $ \Re(t) \geq 0, |\Im(t)| < \Delta $, using compactness and analytic continuation.

Experimental results

Research questions

• RQ1 How are the coefficients of the genus expansion in the large-$ N $ limit of Hermitian random matrix partition functions related to combinatorial enumeration?
• RQ2 What is the functional structure of the genus expansion coefficients $ e_g(t) $, and can they be expressed in terms of a single algebraic function?
• RQ3 How does the shock formation (caustic) in the solution of the inviscid Burgers equation relate to the singularities of the generating functions?
• RQ4 What is the precise connection between the double-scaling limit of the genus expansion and the asymptotic coefficients of the first Painlevé transcendent?
• RQ5 Can the asymptotic expansions of the partition function and recurrence coefficients be uniformly valid in a complex domain extending to infinity in the coupling parameter $ t $?

Key findings

• For $ g \geq 2 $, the genus expansion coefficients $ e_g(t) $ are rational functions of $ z_0(-t) $, the generating function for generalized Catalan numbers.
• The function $ z_0(t) $ solves the inviscid Burgers equation with specific initial data, and its shock formation (caustic) corresponds exactly to the poles of the rational generating functions $ e_g(t) $.
• The asymptotic coefficients of the first Painlevé transcendent in the double-scaling limit are expressed explicitly via the partial fractions expansion of the rational functions $ e_g(t) $.
• The asymptotic expansion of the one-point correlation function $ \rho_1^{(N)}(t, \lambda) $ is uniformly valid near $ t = \infty $, with coefficients analytic in $ t $ for $ z_0(-t) \in [0, z_0^*) $.
• The genus expansion $ \log Z_N(t) = N^2 e_0(t) + e_1(t) + \cdots $ and the recurrence coefficients $ b_{N,N}^2(t) $ admit full asymptotic expansions uniformly valid in a semi-infinite strip $ \Re(t) \geq 0, |\Im(t)| < \Delta $.
• The expansions are uniformly convergent with error bounded by $ C N^{-2h-2} $, and their derivatives may be computed term-by-term in the specified domain.