Troisième rencontre de probabilités intégrables

La troisième rencontre de probabilités intégrables aura lieu le vendredi 14 novembre 2025 à l'Institut Henri Poincaré.

Un mini-cours sera donné par Maurice Duits (KTH Stockholm) et un exposé invité par Ivan Dornic (SPEC, CEA Paris-Saclay). Des exposés courts devraient compléter le programme.

Titres et résumés

Mini-cours (2x1h) de Maurice Duits (KTH Stockholm) :

The Gamma-disordered Aztec diamond

The Aztec diamond, under the uniform measure on domino tilings, is one of the classic gems of integrable probability. Its special structure makes it exactly solvable, and over the years this has led to a wealth of remarkable results. Even for more elaborate weightings—such as doubly periodic ones—the model remains tractable and has attracted much recent interest.

In contrast, once the underlying weight structure is disordered, exact solvability typically breaks down. Despite intriguing predictions from physics, rigorous mathematical results in this direction remain scarce.

In this mini-course, I will present a particular random weighting of the Aztec diamond, which can be viewed as a distinguished example of a dimer model in a random environment. Despite the disorder, the model retains enough integrable structure to permit exact computations of key quantities. A central feature is its remarkable connection to directed polymers.

The lectures are designed for a broad audience, and no prior expertise in tilings or integrable probability will be assumed.

Exposé invité (1h) de Ivan Dornic (SPEC, CEA Paris-Saclay) :

Geometry of the Ising persistence problem and  the universal Bonnet-Manin Painlevé VI distribution

(This is a joint work with R. Conte.)
 
Around 1995, Derrida, Hakim, and Pasquier have obtained an extraordinary exact analytical expression for a continuous family of persistence exponents in one of the simplest model of nonequilibrium statistical physics, the Ising model in one space dimension evolving with zero-temperature Glauber dynamics. 
In particular for symmetric random initial conditions, the corresponding value 3/16 was shown in 2018 by Poplavskyi and Schehr to be universal, in the sense that it appears in several a priori unrelated problems, for instance the probability that the random Kac polynomial has no zeroes on the unit interval in the large-degree limit. 
Our main theorem shows that the Ising persistence problem is governed by a new probability law, characterized by a particular sixth Painlevé function (PVI) which defines a global solution
to the Gauss-Codazzi equations in ordinary Euclidean three-dimensional space,  the persistence exponent itself being given by the solution of a connexion problem à la Jimbo that we resolve here thanks to the Borodin-Okounkov formula. 
It also happens that the same PVI also appears in two other problems: one of classical differential geometry considered and solved by Bonnet in 1867, another one of algebraic geometry studied by Manin in 1998.
All the various geometric quantities of these Bonnet surfaces correspond to universal probability distributions which may be thought as extrapolations at the PVI level of the famous Painlevé II Tracy-Widom distributions.