Random Media and Random Structures in Lima

The box-ball system is a transport cellular automaton of balls on the integers introduced by Takahashi and Satsuma in 1990 as a discrete analogue of the KdV equation. The dynamics conserves solitons, solitary waves travelling at a speed proportional to its size and conserve shape and speed even after colliding with other solitons. A ball configuration has soliton components that are fully conserved by the dynamics, which is indeed proven to be a hierarchical translation of those components. A random initial ball configuration with translation invariant distribution and independent components is invariant for the dynamics. The soliton decomposition of a product measure with density less than 1/2 has independent components, and those are explicitly described. In this case, the decomposition is related to the tree induced by the random walk with increments given by the ball configuration. In the translation invariant case, the asymptotic speeds of solitons are computed and shown to satisfy a universal system of linear equations.
The course is based on joint work with Chi Nguyen, Leo Rolla, Minmin Wang and Davide Gabrielli. arXiv 1806.02798, 1812.02437 and 1906.06405.

I will give an introduction to the discrete time, discrete space loop measure and corresponding loop soup, and give relations between this measure and critical models in statistical physics. I will emphasize two models: the Gaussian free field and the loop-erased random walk. I will then discuss results in the last few years on the loop-erased walk that I have proved with a number of co-authors.

This talk is based on a joint paper with Alexandre Gaudillière and Paolo Milanesi (Université Aix Marseille).
We consider a Glauber dynamics associated with the Ising model on a large two-dimensional box with minus boundary conditions and in the limit of a vanishing positive external magnetic field. The volume of this box increases quadratically in the inverse of the magnetic field. We show that at subcritical temperature and for a large class of starting measures, the system rapidly relaxes to some metastable equilibrium before making a transition at an asymptotically exponential random time towards equilibrium. This result is obtained by bounding from above the local relaxation times towards metastable and stable equilibria. This makes possible to give a pathwise description of such a transition, to control the asymptotic behaviour of the mixing time in terms of soft capacities and to give estimates of these capacities.
Reference: Alexandre Gaudillière, Paolo Milanesi, Maria E. Vares. Asymptotic exponential law for the transition time to equilibrium of the metastable kinetic Ising model with vanishing magnetic field (submitted), https://arxiv.org/abs/1809.07044

We consider a continuous time random walk on the rooted binary tree of depth n with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by 2ⁿ⁺¹n and then centered by (log2)n-log n, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a randomly shifted Gumbel random variable, where the law of the shift is characterized as the unique solution to a specific distributional equation. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian Free Field on the tree. Joint work with Aser Cortines and Oren Louidor.

In a joint work with Marcelo R. Hilário and Augusto Teixeira, we investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole. The asymptotic behavior is expected to depend on the density ρ in [0, 1] of the underlying SSEP.
Our first result is a law of large numbers (LLN) for the random walker for all densities ρ except for at most two values ρ_- and ρ₊ in [0, 1], where the speed (as a function of the density) possibly jumps from, or to, 0.
Second, we prove that, for any density corresponding to a non-zero speed regime, the fluctuations are diffusive and a Central Limit Theorem holds. For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed.
Our main results extend to environments given by a family of independent simple symmetric random walks in equilibrium.

We review some old and new results on random walk among a Poisson system of moving traps on the lattice. We start with explaining some results on the survival probability of such a walk and then present some results on the respective path behavior of the walk conditioned on survival.

Branching Brownian motion is a classical process in probability and a member of a wider class of log-correlated fields. It is also a special and very particular case of a larger class of Gaussian processes indexed by trees that can be parametrised by a covariance (or "speed") function A. The behaviour of the process undergoes a very severe phase transition at the special function A (x ) = x, which corresponds to standard BBM.
If A (x ) < x on (0,1), then the extremes are almost as in the iid case and the order of the maximum is √2(t - ¼ln(t )). For BBM, Bramson the logarithmic correction is three times bigger, and if A (x ) > x, for some x ∊ (0,1), then both the linear and the logarithmic term change. While the change of the linear term is continuous in A, the log-corrections change in a very discontinuous way. In this talk we look more closely at how this transition happens by considering speed functions depending on time.We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard BBM.This is based on joint work with Lisa Hartung.

We will review recent progress in understanding the phenomenon of anomalous slow diffusion induced by disorder. We want to focus on the most central class of such examples, i.e. random walks on critical random structures, for instance critical percolation clusters. This question is also known as “the ant in the labyrinth” problem, since Pierre Gilles de Gennes in the 70’s. Joint work with Alexander Fribergh and Gérard Ben Arous.

We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time aymptotic of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure were we determine anomalous behavior of the corresponding return probability and the Green function of the random walk in random layered conductance.
This is joint work with RYOKI FUKUSHIMA RIMS Kyoto.