Schedule 

We consider a dynamical version of frst-passage percolation on the d-dimensional integer lattice with i.i.d. edge weights, where edge weights are resampled independently in time. Let T(n) denote the passage time from the origin to the site n steps along the frst coordinate axis at time t = 0, and let µ(t) denote the expected overlap between the time minimizing paths at time 0 and t > 0. We show that a subdifusive behaviour of T(n) is equivalent to a chaotic behavior of the time minimizing paths, manifested in that µ(t) = o(n). Known bounds for Var(T(n)) thus imply that indeed µ(t) = o(n) for t > 0. As a consequence we show that there are many almost disjoint paths with almost optimal passage time. This gives evidence that earlier work by Sourav Chatterjee for certain Gaussian disordered systems refects a more general principle.

Consider a nonlinear wave equation with a double-well potential. How often does the field jump from one potential well to the other? We answer this question for a cubic equation and random initial data sampled from its invariant measure, the \phi^4 quantum field theory.

This low-temperature asymptotic behaviour is known as an Eyring--Kramers law. We outline how to derive it from the \phi^4 measure. This result is already known in one and two space dimensions; this presentation relates to work (with Nikolay Barashkov, still unpublished) on the three-dimensional case.

Dimer model is one of the simplest but also most intriguing models of statistical mechanics. It is typically studied through its height function, which turns the dimer model into a model of random surfaces. The main question is its large scale behaviour. A remarkable conjecture of Kenyon and Okunkov predicts that the large scale behaviour is in great generality described by the Gaussian free field. This conjecture was proved by Kenyon in the case of Temperleyan boundary conditions. We generalized this result to the piecewise Temperleyan and simply connected domains. As a byproduct, we showed that the a pair of multiple SLE_8 reduces to a more standard SLE_8(ρ) conditional on the hitting point. This decomposition can also be generated to a more general setting, for example, global 2-SLE and intermediate SLE. This talk is based joint works with Nathana ̈el Berestycki, Yu Feng and Hao Wu.

Propagation and generation of "chaos" is an important ingredient for rigorous control of applicability of kinetic theory.  Chaos can here be understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system.  In this talk based on a joint work with Aleksis Vuoksenmaa, we consider the stochastic Kac model with random velocity exchange.  In earlier pioneering works by Janvresse, Carlen, Carvalho and Loss, Mischler and Mouhot, properties such as spectral gap and propagation of chaos starting from certain permutation invariant states have been determined.  Here, we focus on generation of chaos, in the sense of approach to statistical independence and permutation invariance.  We set up suitable random variables and propose methods to control the evolution of their cumulants.

To connect probabilistic lattice models to conformal field theories (CFT), recent works have introduced a novel definition of local fields of the lattice models. In this picture, local fields are probabilistically concrete: they are built from random variables of the model. The key insight is that discrete complex analysis ideas allow to equip the space of local fields with the main algebraic structure of a CFT: a representation of the Virasoro algebra.

In this talk, we will illustrate this approach to CFT with a concrete model: the discrete gaussian free field (DGFF). We construct the space of local fields of polynomials in the gradient of the DGFF and, as a first main result, we prove that it is isomorphic to the Fock space of the free boson CFT. This constitutes the first instance of a one-to-one correspondence between the local fields of a lattice model and those of a CFT. Our second main result concerns scaling limits of correlation functions. We show that, when correlation functions of local fields of the DGFF are renormalized properly —in a way dictated by the algebraic structure—, they converge to the correlation functions of the free boson CFT.

I discuss relations between quantum field theory, soliton equations, and quantum many-body systems of Calogero-Moser type. My first example is a second quantization of the elliptic Calogero-Sutherland system based on a conformal field theory on the torus; in recent work together with Berntson and Lenells, we found that this second quantization defines, at the same time, a quantum version of a soliton equation called non-chiral intermediate long-wave equation. My second example is a research project of Simon Ruijsenaars between 1980-86 which led him to the relativistic generalization of the elliptic Calogero-Sutherland system now known under his name; I argue that the first examples is a non-relativistic limit of the quantum sine-Gordon equation/massive Thirring model which Simon Ruijsenaars intended to construct in his project. 

Consider the Lorentz mirror model on the 2d lattice: at each lattice site, independently place a mirror at 45 degrees to the lattice with some probability p. The orientation of the mirror is chosen independently, say north-west with probability 0<q<1. Loops can then be formed which bounce off the mirrors, or pass straight through lattice sites with no mirror. What is the probability that the loop through some given edge is infinite? For p=1 it is zero, but for 0<p<1 the problem is open.

We study this model where we re-weigh the measure by n^#loops, n>0. We discuss a form of breaking of translation invariance, where for n large, the almost all the loops are trivial loops surrounding black faces, or trivial loops surrounding the white faces. We can see that the method applied also works for a model of loops coming from O(n)-invariant quantum spin chains, where the breaking of translation invariance is known as dimerisation.

In the talk I will report on a recent work on random lozenge tiling of the hexagon with q-Racah weight, introduced by Borodin-Gorin-Rains. This model has received quite some attention in recent years. Based on loop equation techniques, the limit shape and Gaussian free field (GFF) fluctuations have been established. We revisit the model from a q-orthogonal polynomials' perspective.  We give simple proofs for the limit shape and its fluctuations by analysing the recurrence coefficients of the q-orthogonal polynomials (and their time evolution). The extra benefit of this approach is that the conformal structure, and the homeomorphism for the GFF fluctuations, can be simply read of from the asymptotic behavior of the recurrence coefficients. Moreover, we show that the complex structure is related to the complex Burgers equation. This sheds new light on the variational problem for the model. This talk is based on a joint work with Maurice Duits and Erik Duse.