In Activated Random Walk (ARW) on a finite graph a collection of two types of particles (A and S) evolve according to the following dynamics. Type A particles perform independent continuous-time simple random walks with rate 1 jumps. A fixed vertex acts as a "sink", where particles are removed, if they visit it. When an A particle is alone on a vertex, it changes to an S particle at rate lambda, the parameter of the model. When an A particle visits a vertex occupied by an S particle, the latter instantaneously turns into an A particle. Assuming an initial state of only S particles, at discrete time steps a new A particle is added, and the system is allowed to stabilise to a new state with all S particles. We show that on the complete graph with N vertices, under the stationary distribution, the number of particles in the system is of the form  rho(lambda) N + a(lambda) sqrt(N log N) (1+o(1)), with simple explicit functions rho and a. We compare these results to computations on the 2D Abelian sandpile model, where the exact form of the analogous lower order correction term is still open.

(Joint works with C. Moench & L. Taggi and M.W. Elvidge)

Consider the set of connected subgraphs of Zd with no cycle which contain the origin and have exactly n edges. A lattice tree is the random graph obtained by sampling uniformly from the set just described. When d>8, the limit as n goes to infinite of lattice trees is the super Brownian motion.

In this talk I will present the main ideas behind the proof that the scaling limit of the random walk on lattice trees scale to the Brownian motion on the Super Brownian motion.

Joint work with G. Ben Arous, A. Fribergh, M. Holmes and E. Perkins

The talk concerns recent developments in the mathematical study of critical phenomena. The focus will be certain non-planar percolation models which exhibit long-range correlations. Although seemingly difficult to study at first sight, much progress has been made on our understanding of their critical and near-critical behaviour in recent times. I will give a gentle introduction to one of these models.

At low-temperatures, the mixing time of the Ising Glauber dynamics is exponentially slow due to a bottleneck between the mostly plus and mostly minus phases. One may then ask if in some sense this is the “only real” obstruction to fast mixing. While true on the complete graph, on graphs with non-trivial geometry, more local obstructions can slow down mixing even when restricted to the majority plus phase. For instance, near zero magnetizations, the Ising model on a random d-regular graph has a glassy energy landscape. The question of rapid phase ordering asks whether a small amount of bias towards plus in the initialization is enough to avoid the glassiness and lead to rapid mixing to the majority-plus phase. We present recent progress on this question joint with Allan Sly and Youngtak Sohn.

The shattering phase in p-spin models was first proposed in a seminl work of Kirkpatrick-Thirumalai '87. We discuss this phase and its existence in spherical and Ising p-spin models. We  will try to give a compelete sketch of an elementary proof of ethe existence of this phase in Ising p-spin models.

One of the fascinating phenomena in spin glasses is the dramatic change in behavior between the high- and low-temperature regimes. In this talk, we focus on spin glasses near the critical temperature threshold and present results on the fluctuations of the free energy. We examine two variants of the famous Sherrington-Kirkpatrick (SK) model. For the bipartite spherical SK (BSSK) model, we show that the free energy fluctuations at critical temperature converge to a sum of Gaussian and Tracy-Widom distributions, interpolating between known results in the high- and low-temperature regimes. For the multi-species SK (MSK) model (with a positive semi-definite variance profile), we provide a sharper variance bound than previously known. We will highlight the key techniques used for analyzing each model, which differ significantly, as BSSK has continuous spins, while MSK has discrete spins. The talk is based on joint work with E. Collins-Woodfin.

The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. We give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, corresponding to a lower bound on the geometric entanglement of random quantum states. The proof is based on spin-glass methods, the Kac—Rice formula, and recent progress coming from random matrices. Joint work with Stéphane Dartois. ​

Matrix denoising is central to signal processing and machine learning. Its analysis when the matrix to infer has a factorized structure with a rank growing proportionally to its dimension remains a challenge, except when it is rotationally invariant. In this case the information theoretic limits and a Bayes-optimal denoising algorithm, called the rotational invariant estimator, are known. Beyond this setting few results can be found. The reason is that the model is not a usual spin system because of the growing rank dimension, nor a matrix model due to the lack of rotation symmetry, but rather a hybrid between the two. In this talk we make progress towards the understanding of Bayesian matrix denoising when the hidden signal is a factored matrix that is not rotationally invariant. Monte Carlo simulations suggest the existence of a denoising-factorisation transition separating a phase where denoising using the rotational invariant estimator remains Bayes-optimal due to universality properties of the same nature as in random matrix theory, from one where universality breaks down and better denoising is possible by exploiting the signal's prior and factorised structure, though algorithmically hard. We also argue that it is only beyond the transition that factorisation, i.e., estimating X itself, becomes possible up to sign and permutation ambiguities. On the theoretical side, we combine mean-field techniques in an interpretable multiscale fashion in order to access the minimum mean-square error and mutual information. Using numerical insights, we then delimit the portion of the phase diagram where this mean-field theory is reliable, and correct it using universality when it is not. Our ansatz matches well the numerics when accounting for finite size effects. This is joint work with Jean Barbier, Francesco Camilli, and Koki Okajima.

The elastic random manifold serves as a paradigmatic example of an elastic interface suspended in a quenched disordered medium, with random polymers models being the most well-studied special case. Work in the physics community has suggested that the behavior of these models in high dimensions should enter a number of distinct phases, depending on the temperature, and structure of the disorder, with the depinning transition being the most studied.

We confirm a number of predictions for these phases, including existence, rigorously. The central result is a formula for the free energy in the high-dimensional limit, from which we are also able to obtain the behavior of a number of other statistics. The key tool in our computation involves adapting the multi-species synchronization method of Panchenko.

This is a joint work with Gerard Ben Arous.

We consider the problem of efficiently optimizing random (spherical or Ising) perceptron models with general bounded Lipschitz activation. We focus on a class of algorithms with Lipschitz dependence on the disorder: this includes gradient descent, Langevin dynamics, approximate message passing, and any constant-order method on dimension-free time-scales. Our main result exactly characterizes the optimal value ALG such algorithms can attain in terms of a one-dimensional stochastic control problem. Qualitatively, ALG is the largest value whose level set contains a certain "dense solution cluster." Quantitatively, this characterization yields both improved algorithms and hardness results for a variety of asymptotic regimes, which are sharp up to absolute constant factors. Joint work (in progress) with Mark Sellke and Nike Sun.

 We consider the questions of sampling and finding solutions of prescribed energy in mean field spin glasses. We will show the existence of a regime of temperatures where the Gibbs measure exhibits shattering, and discuss the algorithmic implications on approximate sampling, and the easier task of finding solutions of typical energy. In particular, we show for the latter task that there exists an exponentially small subset of the configuration space with the property that any stable algorithm is very likely to land in this rare set, provided that it succeeds at finding a solution at all.

Random walks on particle systems, such as the simple exclusion process (SEP), have undergone notable developments in the last decade. In such models, say on the $d$-dimensional lattice, the law of the (discrete-time) movement of the walker depends on whether its current location is occupied or not by a particle of the environment.

We will focus on the case d=1, when occupied sites have a drift to the right, and empty sites a drift to the left. The parameter of interest will be the density of particles: when it is large (resp. small), the random walk has a positive (resp. negative) speed. 

Since the SEP is conservative and mixes slowly, a natural question is whether there are strong trapping effects, as in the more classical setting of static environments. Could it be for instance that for a non-empty interval of intermediate densities, the random walk has zero speed? We give a negative answer, showing that the speed is strictly increasing with the density.

The proof uses a comparison with a finite-range model (via renormalisation), and an original coupling to circumvent the bad mixing properties of the SEP. 

This is joint work with Daniel Kious and Pierre-François Rodriguez.