A PDE Approach to the Combinatorics of the Full Map Enumeration Problem: Exact Solutions and their Universal Character

Maps are polygonal cellular networks on Riemann surfaces. This paper completes a program of constructing closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the tiling polygons. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable {\it unwinding identity} in terms of the Appell polynomials generated by Bessel functions. Our treatment, based on random matrix theory and Riemann-Hilbert problems for orthogonal polynomials reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis as well as directly related to universality results for random matrix spectra as described by Deift, Kriecherbauer, McLaughlin, Venakides and Zhou

The spectrum of an asymmetric annihilation process

International audienceIn recent work on nonequilibrium statistical physics, a certain Markovian exclusion model called an asymmetric annihilation process was studied by Ayyer and Mallick. In it they gave a precise conjecture for the eigenvalues (along with the multiplicities) of the transition matrix. They further conjectured that to each eigenvalue, there corresponds only one eigenvector. We prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization. In addition, we outline a derivation of the partition function in the generalized model, which also reduces to the one obtained by Ayyer and Mallick in the original model.Dans un travail récent sur la physique statistique hors équilibre, un certain modèle d'exclusion Markovien appelé "processus d'annihilation asymétrique'' a été étudié par Ayyer et Mallick. Dans ce document, ils ont donné une conjecture précise pour les valeurs propres (avec les multiplicités) de la matrice stochastique. Ils ont en outre supposé que, pour chaque valeur propre, correspond un seul vecteur propre. Nous prouvons la première de ces conjectures en généralisant la matrice originale de Markov par l'introduction de paramètres supplémentaires, calculant explicitement ses valeurs propres, et en montrant que la nouvelle matrice se réduit à l'originale par une spécialisation appropriée. En outre, nous présentons un calcul de la fonction de partition dans le modèle généralisé, ce qui réduit également à celle obtenue par Ayyer et Mallick dans le modèle original

Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide

We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems - which includes exclusion processes - focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.Comment: 127 pages, 31 figures, invited topical review for J. Phys. A (uses IOP class file

Stieltjes moment sequences for pattern-avoiding permutations

International audienceA small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We show that the generating functions of the sequences $\, Av(1234)$ and $\, Av(12345)$ correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian \, _2F_1 hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a \, _2F_1 hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence $Av(123\ldots k)$ is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with $k-1$ unit steps in random directions. Finally, we study the challenging case of the $Av(1324)$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant