Random Media and Random Structures in Lima

We study the stationary distribution of the (spread-out) d-dimensional contact process from the point of view of site percolation. In this process, vertices of Z^d can be healthy (state 0) or infected (state 1). With rate one infected individuals recover, and with rate λ they transmit the infection to some other vertex chosen uniformly within a ball of radius R. The classical phase transition result for this process states that there is a critical value λ_c(R) such that the process has a non-trivial stationary distribution if and only if λ > λ_c(R). In configurations sampled from this stationary distribution, we study nearest-neighbor site percolation of the set of infected sites; the associated percolation threshold is denoted λ_p(R). We prove that λ_p(R) converges to 1/(1-p_c) as R tends to infinity, where p_c is the threshold for Bernoulli site percolation on Z^d. As a consequence, we prove that λ_p(R) > λ_c(R) for large enough R, answering an open question of [Liggett, Steif, AIHP, 2006] in the spread-out case. Joint work with Balázs Ráth.

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.

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.

The KPZ universality class is a broad collection of models, including directed random polymers, particle systems and random interface growth, characterized by a rich and unusual fluctuation behavior which is model independent but depends on the initial data, and is in some cases related to random matrix theory. The KPZ fixed point is a special scaling invariant Markov process which is the conjectural universal limit of all models in the class and contains all its fluctuation behavior. I will introduce and describe this object and how it arises from certain special models in the class. Then I will explain how the KPZ fixed point is connected to a famous completely integrable dispersive PDE, the Kadomtsev-Petviashvili (KP) equation. Surprisingly, this connection also holds for some special solutions of the KPZ equation, the famous SPDE which gives its name to the class.

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.

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.

We will start with discussion of global solutions in the compact (periodic) case where the theory is well understood by now. We'll introduce the main objects: minimizers, shocks, and discuss the hyperbolicity properties of minimizers. We then move to the theory in non-compact setting. We shall discuss results in the 1D case, and in the higher dimensional weak disorder situation. In the 1D case we present the renormalization scheme based on geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We conjecture that the KPZ universality can be explained in terms of statistics of these structures evolving in time.

In the classical finite dimensional setting, a Gaussian multiplicative chaos (GMC) is obtained by tilting an ambient measure by the exponential of a centred Gaussian field indexed by a domain in the Euclidean space. In the two-dimensional setting and when the underlying field is "log-correlated", GMC measures share close connection to the 2D Liouville quantum gravity, which has seen a lot of revived interest in the recent years.
A natural question is to construct a GMC in the infinite dimensional setting, where techniques based on log-correlated fields in finite dimensions are no longer available. In the present context, we consider a GMC on the classical Wiener space, driven by a (mollified) Gaussian space-time white noise. In d ≥ 3, in a previous work with A. Shamov and O. Zeitouni, we showed that the total mass of this GMC, which is directly connected to the (smoothened) Kardar-Parisi-Zhang equation in d ≥ 3, converges for small noise intensity to a well-defined strictly positive random variable, while for larger intensity (i.e. for small temperature) it collapses to zero. We will report on joint work with Yannic Bröker (Münster) where we study the endpoint distribution of a Brownian path under the GMC measure and show that, for low temperature, the endpoint GMC distribution localizes in few spatial islands and produces asymptotically purely atomic states.

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.

This talk is based on joint work with Thomas Mountford and Hao Xue [1]. We consider an anisotropic finite-range bond percolation model on Z². On each horizontal layer H_i = {(x, i) : x ∈ Z} we have edges ⟨(x, i), (y, i)⟩ for 1 ≤ |x − y| ≤ N. There are also vertical edges connecting two nearest neighbor vertices on distinct lines ⟨(x, i), (x, i + 1)⟩ for x, i ∈ Z. On this graph we consider the following anisotropic independent percolation model: horizontal edges are open with probability 1/(2N), while vertical edges are open with probability ε to be suitably tuned as N grows to infinity. The main result tells that if ε = κN^{− 2/5}, then we see a phase transition in κ: there exist positive and finite constants C₁, C₁ so that there is nopercolation if κ < C₁, while percolation occurs for κ > C₂. The question is motivated by a result on the analogously layered ferromagnetic Ising model at mean field critical temperature [2].
[1] T. S. Mountford, M. E. Vares, H. Xue. Critical scaling for an anisotropic percolation system on Z² (arxiv.org/abs/1904.11030)
[2] L. R. Fontes, D. Marchetti, I. Merola, E. Presutti, M. E. Vares. Layered Systems at the Mean Field Critical Temperature. J. Stat. Phys. (2015) 161, 91–123. Mean Field Critical Temperature. J. Stat. Phys. (2015) 161, 91–123.

I'll discuss joint work with Federico Camia and Jianping Jiang (arXiv:1910.12742). The continuum scaling limit of the Ising model in d dimensions at the critical temperature whose magnetic field properly scales to zero with lattice spacing is (or should be) a non-Gaussian generalized random field Φ for d = 2 (and d = 3). This field is (or should be) related to a relativistic quantum field theory with one time and d-1 space coordinates and no particles of mass below some m₁ > 0. To help study 30 year old conjectures/predictions of Zamolodchikov about masses m₂, m₃, ... for d = 2, we study a further scaling limit of Φ which yields a stationary Gaussian process X(t) in the time coordinate t. The covariance function of the X process contains information about the spectrum of particle masses.

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 - ln(t)/4). 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.

Activated random walks is a system of particles which perform random walks and can spontaneously fall asleep, staying put. When an active particle falls on a sleeping one, the sleeping particle becomes active and continues moving. The system displays a phase transition in terms of the density of particles. If the density is small, all particles will eventually sleep forever, while, if the density is high, the system can sustain a positive proportion of active particles. In this talk we describe the critical behavior of the model in the totally asymmetric case.

We investigate the behavior of a lattice free field interacting with a potential near the origin. When the potential is homogeneous, it is known that the system exhibits a first-order phase transition between a localized phase and a delocalized one. We show that adding disorder into the system changes the critical behavior of the system and makes the phase transition quadratic, and we identify the typical behavior of the field close to the phase transition.
Joint work with G. Giacomin.

We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian. These measures are parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, and above a critical density. Each finite cycle of the permutation induces a loop of points. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. The resulting measure satisfies a Markov property and it is Gibbs for the Gaussian Hamiltonian. We show that the point marginal has the same distribution as the boson point process introduced by Shirai-Takahashi (2003) in the subcritical case, and by Tamura-Ito (2007) in the supercritical case.
Joint work with P.A. Ferrari and S. Yuhjtman.

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.

We consider the abelian stochastic sandpile model. In this model, a site is unstable if it contains two or more particles. Each unstable site is toppled at rate 1, sending two of its particles to neighboring sites. The target site is chosen to the left with probability q and to the right with probability 1-q, independently of the other particle. In a joint work with Vladas Sidoravicius, we showed that if the initial average density is less than q(1-q), the system locally fixates almost surely. This was achieved by considering a modified dynamics and introducing an appropriate toppling procedure. In a recent work with Moumanti Podder, we show almost sure fixation when the initial average density is less than 1/2, for every 0 < q < 1. We achieve this new bound by considering a modification of the previous toppling procedure and analyzing the parity of the total number of times each site is visited in previous steps.