Guillaume Chapuy, Maps and tableaux, through loop equations and constellations

Grand generating functions of maps on surfaces of arbitrary genus, with infinitely many variables marking all vertex degrees, can classically be expressed, by means of representation theory of the symmetric group, as infinite sums over integer partitions involving Schur functions. In other words, as some generating functions of Young tableaux. These expressions are important because they are at the core of the structure of KP/2–Toda tau-function of generating functions of maps, yet their combinatorial meaning is totally unclear. I will present a (new?) concrete proof of these expressions that does not rely on representation theory, and is almost solely based on the loop equations. Constellations – some sorts of highly decorated maps – play an important role in the story. If time permits I will also evoke some ramifications of these ideas, that lead to substantial progress towards some hot conjectures in algebraic combinatorics. This is work in progress with Maciek Dołęga (Wrocław).

Linxiao Chen, A positivity bootstrap technique for validating the generating function of loop-decorated maps

Most loop equations originating from map enumeration problems share the feature that the uniqueness of their solution is easy to check. However, for a recently solved model of loop-decorated maps this is no longer true. I will explain a way to tackle this problem by combining (a) a martingale argument used on the recursive construction of these loop-decorated maps, and (b) a positivity bootstrap technique to determine the sign of the so-called spectral density for the resolvent (≈ generating function) of these loop-decorated maps.

Antoine Dahlqvist, Makeenko–Migdal equations and the Yang–Mills measure on the sphere

I will explain how the Makeenko–Migdal equations and the study of a 1D Coulomb gas allow to show the convergence of Wilson loops on the sphere, under the Yang–Mills measure. I shall present the results of a joint work with J. Norris, as well as some further investigations.

Béatrice de Tilière, The Z-Dirac operator and massive Laplacian operators in the Z-invariant Ising model

The subject of this talk is Baxter's Z-invariant Ising model defined on isoradial graphs. We prove that certain key quantities of the model – the partition function, and edge-probabilities in the contour representation – can be explicitly expressed using the Z-massive Laplacian and its inverse, the massive Green function, introduced by Boutillier, dT, Raschel. This establishes a deep relation between classical 2d-models of statistical mechanics: the Ising model, spanning forests and random walk. In order to prove these results, we introduce the Z-Dirac operator; we relate it to the Z-massive Laplacian, extending to the full Z-invariant regime results obtained by Kenyon at the critical point; we then relate the Z-Dirac operator to the Ising model. The proof consists in establishing matrix relations allowing to compare matrix inverses and also, using additional arguments, determinants.

Franck Gabriel, Permutation invariant Lévy processes and application to random walks on symmetric groups

The Schwinger–Dyson (S.D.) equations for Gaussian measure can be obtained either by integration by part or by using Itô calculus. This last method can be generalized to Lévy processes in order to obtain a general theorem for the convergence in probability of matrices-valued Lévy processes (when the size of the matrices tends towards infinity). We will apply these results to random walks, on the symmetric group, which jump uniformly in any given family of conjugacy classes.

Mylène Maïda, On the Douglas–Kazakov phase transition

Yuri Makeenko, Matrix models with singular potentials

I show that the loop equations apply for matrix models with the potentials whose derivative has a cut at the real axis and elaborate on some explicit examples. I consider the proper generalizations of Kazakov's multi-critical potentials, the moments and the Gelfand–Dikii polynomials of the KdV hierarchy.

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