Name: Journée cartes à l'IPhT
Start: 2022-06-24T09:30:00+02:00
End: 2022-06-24T16:15:00+02:00
Location: IPhT

09:30

Café d'accueil

Hall d'entrée

1

Nicolas Curien -- Parking on the infinite binary tree

Let $(A_u : u \in B)$ be i.i.d. non-negative integers that we interpret as car arrivals on the vertices of the full binary tree $B$. Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known that the parking process on $B$ exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We characterize those regimes in terms of the law of $A$ in an explicit way. We also study in detail the critical regime as well as the phase transition which turns out to be "discontinuous".

Joint work with David Aldous, Alice Contat and Olivier Hénard.

11:15

Pause

2

Fedor Levkovich-Maslyuk -- Critical phenomena for dually weighted graphs and spanning forests via matrix models

I will discuss two nontrivial problems of graph counting that are solvable by reduction to matrix models. The first one is summing over planar graphs with weights for both vertices and faces. In this setting one can control the geometry efficiently and obtain in the continuum limit a nontrivial 2d theory of quantum gravity ($R^2$ type). I will compute the disk partition function in this model and show that it interpolates between pure gravity and the regime of almost flat surfaces with a gas of conical defects.

The second problem is counting rooted spanning forests on planar trivalent graphs. Equivalently this corresponds to massive spinless fermions interacting with 2d quantum gravity. I will show how this model can be solved, demonstrate that it interpolates between $c=-2$ and $c=0$ regimes, and explore its phase structure and critical behavior.

12:30

Déjeuner

Buffet dans la bibliothèque

3

Zéphyr Salvy -- Random planar maps decomposed into blocks: a phase study

Maps come with different shapes, such as trees or triangulations with many more edges. Many classes of maps have been enumerated (2-connected maps, trees, quadrangulations...), notably by Tutte, and a phenomenon of universality has been demonstrated: for the majority of them, the number of elements of size $n$ in the class has an asymptotic of the form $\kappa\, \rho^{-n} \, n^{-5/2}$, for a certain $\kappa$ and a certain $\rho$. Nevertheless, there are classes of "degenerate" maps whose behaviour is similar to that of trees, and whose number of elements of size $n$ has an asymptotic of the form $\kappa\, \rho^{-n} \, n^{-3/2}$, as for example outerplanar maps. This dichotomy of behaviour is not only observed for enumeration, but also for metrics. Indeed, in the "tree" case, the distance between two random vertices is in $\sqrt{n}$, against $n^{1/4}$ for uniform planar maps of size $n$. This work focuses on what happens between these two very different regimes. We highlight a model depending on a parameter $u \in \mathbb{R}^*_+$ which exhibits the expected behaviours, and a transition between the two: depending on the position of $u$ with respect to $u_C$, the behaviour is that of one or the other universality class. More precisely, we observe a "subcritical" regime where the scaling limit of the maps is the Brownian map, a "supercritical" regime where it is the Brownian tree and finally a critical regime where it is the $3/2$ stable tree.

15:00

Pause

4

David Keating -- $k$-tilings of the Aztec diamond

We will study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond. We assign a weight to each $k$-tiling, depending on the number of "interactions" between the dominos of the different tilings. We will compute the generating polynomials of the $k$-tilings by showing that they can be seen as the partition function of an integrable colored vertex model. These partition functions are related to the LLT polynomials of Lascoux, Leclerc, and Thibon. We will then prove some combinatorial results about $k$-tilings in certain limits of the interaction strength.

Journée cartes à l'IPhT

amphithéâtre Bloch

IPhT