# Schedule

All talks to take place in room B3.03 in the mathematics building

#### Monday 7 April

9:20 Welcome

9:30-10:30 **Miklós Abért**, *Benjamini-Schramm convergence, graph polynomials and entropy of cellular automata*

coffee

11:00-12:00 **Elena Pulvirenti**, *Phase transitions and coarse graining for a system of particles in the continuum*

We consider a system of particles in R^d interacting via a reasonable potential with both long and short range contributions and prove rigorously the existence of a liquid-vapor branch in the phase diagram of fluids. The model we consider is a variant of the model introduced by Lebowitz, Mazel and Presutti (1999), obtained by adding a hard core interaction to the original Kac potential interaction, the first acting on a scale of order 1 with respect to the Kac parameter. We prove perturbatively that if the hard core radius R is sufficiently small, then the liquid-vapor phase transition proved for the LMP model is essentially unaffected. Hence, we prove existence of two different Gibbs measures corresponding to the two phases. This is a joint work with Errico Presutti and Dimitrios Tsagkarogiannis.

lunch

14:00-15:00 **Nathanaël Berestycki**, *Ricci curvature and mixing times on the permutation group*

I will discuss a discrete notion of Ricci curvature, introduced by Ollivier, in the context of the permutation group equipped with the generating set of transpositions. In particular I will show that a phase transition takes place between zero and nonzero curvature asymptotically when the random walk is run for time c n, according to whether c<1/2 or c >1/2. This answers a question of Ollivier. A surprising application of this result is an easy and "geometric" proof of the Diaconis-Shahshahani result that random transpositions mix in time t= (1/2) n log n. In fact, the proof is general and works for random walks on the permutation group based on any conjugacy class of bounded (or even moderate) size.

This is joint work with Bati Sengul.

coffee/tea

15:30-16:30 **Alexey Gladkich**, *The cycle structure in random Mallows permutation*

The Mallows model is a probability measure on permutations in S_n in which the probability of a permutation pi is proportional to q^{inv(pi)}, where inv(pi) denotes the number of inversions in pi and 0<q<1 is a parameter of the model. The model is an example of a class of distributions called spatial random permutations in which the distribution is biased to be close to the identity in a certain underlying geometry. We study the cycle structure of a permutation sampled from the Mallows model, partially addressing a question of Borodin, Diaconis and Fulman. Our main result is that the expected length of the cycle containing a given point is of order min(1/(1-q)^2, n). In contrast, the expected length of a uniformly chosen cycle is of order min(1/(1-q),n / log(n)). We also overview some related results and conjectures about spatial random permutations. No prior knowledge of random permutations will be assumed.

Joint work with Ron Peled.

17:00 Welcome drink

#### Tuesday 8 April

9:30-10:30 **Balint Tóth**, *Superdiffusive CLT for periodic Lorentz gas in the Boltzmann-Grad limit*

We prove central limit theorem under superdiffusive scaling $\sqrt{t \log t}$ for the displacement of particle in the $Z^d$-based periodic Lorentz gas, in the Boltzmann-Grad limit. The result holds in any dimension. This is joint work with Jens Marklof (Bristol).

coffee

11:00-12:00 **Gady Kozma**, *Random walks in divergence-free random environment*

For this class of non-Markovian processes, we show conditions under which they satisfy a central limit theorem. All notions will be explained in the talk, including the theory of Kipnis & Varadhan which plays a central role. Joint work with Balint Tóth.

lunch

14:00-15:00 **June Huh**, *Rota's conjecture and positivity of algebraic cycles in toric varieties* + board pictures 1 + board pictures 2

Rota's conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. I will outline a proof for realizable matroids using techniques of algebraic geometry. The same approach to the conjecture in the general case (for possibly non-realizable matroids) leads to several intriguing questions on higher codimension algebraic cycles in the toric variety associated to the permutohedron.

coffee/tea

15:30-16:30 **Christian Scullard**, *The polynomial method for percolation and the Potts model*

#### Wednesday 9 April

9:30-10:30 **Roberto Fernández**, *Convergence of cluster expansions: A review of the main strategies and their relations*

coffee

11:00-12:00 **Stephen Tate**, *The combinatorics of the virial expansion*

This talk addresses a challenge posed by Ducharme Labelle and Leroux asking for a combinatorial proof of identities arising from Mayer's theory of cluster and virial expansions. This was answered in the case of connected graphs in the cluster expansion by Bernardi (08). A solution to the two connected virial expansion case is presented. The connection between Bernardi's work and the Penrose construction is presented in order to motivate how a similar approach could be achieved for the virial expansion.

12:00-13:00 **Aldo Procacci**, *Witness trees in the Moser-Tardos algorithmic Lovász Local Lemma and Penrose trees in the hard core lattice gas*

We point out a close connection between the Moser-Tardos algorithmic version of the Lov\'asz Local Lemma, a central tool in probabilistic combinatorics, and the cluster expansion of the hard core lattice gas in statistical mechanics. We show that the notion of witness trees given by Moser and Tardos is essentially coincident with that of Penrose trees in the Cluster expansion scheme of the hard core gas. Such an identification implies that the Moser Tardos algorithm is successful in a polynomial time if the Cluster expansion converges.

lunch

#### Thursday 10 April

9:30-10:30 **Peter Mörters**, *The universal shape of Bose-Einstein condensates*

coffee

11:00-12:00 **Márton Balázs**, *A new connection between irreversible random walks and electric networks*

There is a well-known analogy between reversible Markov chains and electric networks: the probability of reaching one state before another agrees with voltages in a corresponding network of resistances, and the electric current also has a probabilistic interpretation. Such analogies can be used to prove a variety of theorems regarding transience-recurrence, commute times, cover times. The electric counterpart is very simple, consists of resistors only. These simple components behave in a symmetric fashion, that's why the analogy

only works for reversible chains.

We found the electric component that allows to extend the above analogy from reversible Markov chains to irreversible ones. I will describe this new component, show how the analogy works, demonstrate some arguments that can be saved from the reversible case and some which fail, at least directly. I will outline ongoing investigations towards generalisations of statements from the reversible case that would still hold for the irreversible chain - this is still work in progress, joint with Aron Folly.

lunch

14:00-15:00 **Jan Swart**,* Random 3-colorings and a phase transition for the 3-state Potts antiferromagnet on a class of planar lattices*

In antiferromagnetic Potts models, sites in a lattice prefer to be of a different type than their neighbors. In particular, zero-temperature Gibbs states of such a model are random colourings of the lattice. For each lattice, there is believed to be a critical number $q_c$ such that the antiferromagnetic $q$-state Potts model is disordered at zero temperature for $q>q_c$ and exhibits a phase transition between order and disorder at some positive temperature for $q<q_c$. For the square lattice $\Z^2$, it is believed that $q_c=3$ and in fact the 3-state model is critical at zero temperature. We describe a class of periodic planar bipartite graphs for which we can prove that, in contrast to $\Z^2$, the 3-state antiferromagnetic Potts model exhibits long-range order for sufficiently small temperatures, and speculate on the possible reasons for this different behaviour. This is joint work with A.D. Sokal (New York) and R. Kotecký (Warwick).

coffee/tea

15:30-16:30 **Ohad Feldheim**, *Rigidity of 3-colorings of high dimensional discrete torus* + board pictures

We prove that a uniformly chosen proper cologing of $Z^d / NZ^d$ with 3 colors, has very rigid structure when the dimension d is sufficiently high. When $N$ goes to infinity, either the even or the odd is almost surely nearly monochromatic. This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in statistical mechanics. The proof involves results about graph homomorphisms and various combinatorial methods, and follows a topological intuition. Joint work with Ron Peled.

18:00 Conference dinner (mathematics common room)

#### Friday 11 April

9:30-10:30 **Christoph Temmel**, *The non-physical singularity of the one-dimensional hard-sphere model*

coffee

11:00-12:00 **Yinon Spinka**, *The loop O(n) model* + board pictures

A loop configuration on the hexagonal (honeycomb) lattice is a finite subgraph of the lattice in which every vertex has degree 0 or 2, so that every connected component is isomorphic to a cycle. The loop O(n) model on the hexagonal lattice is a random loop configuration, where the probability of a loop configuration is proportional to x^(#edges) n^(#loops) and x,n>0 are parameters called the edge-weight and loop-weight. We show that for sufficiently large n, the probability that the origin is surrounded by a loop of length k decays exponentially in k. In this same region of parameters, we also show a phase transition from a disordered phase to an ordered phase. No prior knowledge in statistical mechanics will be assumed. All notions will be explained. Joint work with Hugo Duminil-Copin, Ron Peled and Wojciech Samotij.

12:00-13:00 **Adrian Tanasa**, *Some combinatorics of random tensor models*

Random tensor models, seen as quantum field theoretical models, represent a natural generalization of the celebrated 2-dimensional matrix models. These matrix models are known to be connected to 2-dimensional quantum gravity, and one of the main results of their study is that their perturbative series can be reorganized in powers of 1/N (N being the matrix size). The leading order in this expansion is given by planar graphs (which are dual to triangulations of the 2-dimensional sphere S^2).

In this talk I will present such a 1/N asymptotic expansion for some particular class of 3-dimensional tensor models (called multi-orientable models). The leading order (and hence the dominant graphs, dual to particular triangulations of the three-dimensional sphere S^3), the next-to-leading order and finally some considerations on the combinatorics of the general term of this asymptotic expansion will be given.

lunch