In the 40s R. Feynman invented a simple model of electron motion, which is now known as Feynman's checkers. This model is also known as the one-dimensional quantum walk or the imaginary temperature Ising model. In Feynman's checkers, a checker moves on a checkerboard by simple rules, and the result describes the quantum-mechanical behavior of an electron.
We solve mathematically a problem by R. Feynman from 1965, which was to prove that the model reproduces the usual quantum-mechanical free-particle kernel for large time, small average velocity, and small lattice step. We compute the small-lattice-step and the large-time limits, justifying heuristic derivations by J. Narlikar from 1972 and by A.Ambainis et al. from 2001. The main tools are the Fourier transform and the stationary phase method.
A more detailed description of the model can be found in Skopenkov M.& Ustinov A. Feynman checkers: towards algorithmic quantum theory. (2020) https://arxiv.org/abs/2007.12879[-]

Maps decorated by the Ising model are a remarkable instance of a model of non-uniform maps with very nice enumerative properties. In this talk, I will first explain how one can obtain a differential equation for the generating function of Ising-decorated cubic maps in arbitrary genus, related to the Kadomtsev--Petviashvili (KP) hierarchy. In particular, this leads to an efficient algorithm to enumerate Ising cubic maps in high genus. I will also present and compare implementations of this algorithm in Maple and SageMath. This is based on a joint work with Mireille Bousquet-Mélou and Baptiste Louf.[-]

Après avoir expliqué la notion de Z-invariance pour les modèles de mécanique statistique, nous introduisons une famille à un paramètre (dépendant du module elliptique) de Laplaciens massiques Z-invariants définis sur les graphes isoradiaux. Nous démontrons une formule explicite pour son inverse, la fonction de Green massique, qui a la propriété remarquable de ne dépendre que de la géométrie locale du graphe. Nous expliquerons les conséquences de ce résultat pour le modèle des forêts couvrantes, en particulier la preuve d'une transition de phase d'ordre 2 avec le modèle des arbre couvrants critiques sur les graphes isoradiaux, introduit par Kenyon. Finalement, nous considérons la courbe spectrale de ce Laplacien massique et montrons qu'il s'agit d'une courbe de Harnack de genre 1.
Il s'agit d'un travail en collaboration avec Cédric Boutillier et Kilian Raschel.[-]

Les chemins du plan confinés dans un quadrant, ou plus généralement dans un cône convexe, ont été beaucoup étudiés ces dernières années, et ont donné lieu à de jolis résultats. Le plus remarquable dit que, pour les chemins à petits pas, la série génératrice est différentiellement finie si et seulement si un certain groupe de transformations rationnelles, construit à partir des pas autorisés, est fini. Les méthodes employées, allant de l'algèbre élémentaire sur les séries formelles à l'analyse complexe, en passant, entre autres, par le calcul formel, sont variées, ce qui participe au charme du sujet. Mais quid des chemins dans un cône non convexe, et, typiquement, des chemins évitant un quadrant ? On étudiera les deux cas les plus naturels (pas NSEO, quadrant négatif ou quadrant Ouest interdit), en esquissant avec optimisme ce que pourrait être une classification pour ce problème.[-]

We propose a platform for finding the global minimum of XY Hamiltonian with polariton graphs. We derive an approximate analytic solution to the spinless complex Ginzburg-Landau equation that describes the density and kinetics of a polariton condensate under incoherent pumping. The analytic expression of the wavefunction is used as the building block for constructing the XY Hamiltonian of two-dimensional polariton graphs. We illustrate examples of the quantum simulator for various classical magnetic phases on some simple lattice geometries: linear, triangular, square.[-]

The two-periodic Aztec diamond is a dimer or random tiling model with three phases, solid, liquid and gas. The dimers form a determinantal point process with a somewhat complicated but explicit correlation kernel. I will discuss in some detail how the Airy point process can be found at the liquid-gas boundary by looking at suitable averages of height function differences. The argument is a rather complicated analysis using the cumulant approach and subtle cancellations. Joint work with Vincent Beffara and Sunil Chhita.[-]

The aim of this talk is to show how basic notions traditionally used in the study of "knotted embeddings in dimensions $3$ and $4$", such as covering spaces and representation theory, can have non-trivial applications in combinatorics and statistical mechanics. For example, we will show that for any finite covering $G'$ of a finite edge-weighted graph $G$, the spanning tree partition function on $G$ divides the spanning tree partition function on $G'$ (in the polynomial ring with variables given by the weights). Setting all the weights equal to $1$, this implies a theorem known since 30 years: the number of spanning trees on $G$ divides the number of spanning trees on $G'$. Other examples of such results will be presented.
Joint work (in progress) with Adrien Kassel.[-]

Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs – one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general theory, highlight the unbounded graphons, and show how they can be used to consistently estimate properties of large sparse networks. This talk will also give an application of these sparse graphons to collaborative filtering on sparse bipartite networks. Talk II, given by Christian, will recast limits of dense graphs in terms of exchangeability and the Aldous Hoover Theorem, and generalize this to obtain sparse graphons and graphexes as limits of subgraph samples from sparse graph sequences. This will provide a dual view of sparse graph limits as processes and random measures, an approach which allows a generalization of many of the well-known results and techniques for dense graph sequences.[-]

Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs – one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general theory, highlight the unbounded graphons, and show how they can be used to consistently estimate properties of large sparse networks. This talk will also give an application of these sparse graphons to collaborative filtering on sparse bipartite networks. Talk II, given by Christian, will recast limits of dense graphs in terms of exchangeability and the Aldous Hoover Theorem, and generalize this to obtain sparse graphons and graphexes as limits of subgraph samples from sparse graph sequences. This will provide a dual view of sparse graph limits as processes and random measures, an approach which allows a generalization of many of the well-known results and techniques for dense graph sequences.[-]