Cyril Roberto

Hamilton Jacobi equations on metric spaces and transport entropy inequalities

We prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton- Jacobi equations on a general metric space. As a first consequence, we show in full gener- ality that the log-Sobolev inequality is equivalent to an hypercontractivity property of the Hamilton-Jacobi semi-group. As a second consequence, we prove that Talagrands transport- entropy inequalities in metric space are characterized in terms of log-Sobolev inequalities restricted to the class of c-convex functions.

Kinetically Constrained Models

Kinetically constrained spin models (KCSM) are interacting particle systems which are intensively studied in physics literature as models for systems undergoing glass or jamming transitions. KCSM leave on discrete lattices and evolve via a Glauber-like dynamics which is reversible w.r.t. a simple product measure. The key feature is that the creation/destruction of a particle at a given site can occur only if the current configuration satisfies proper local constraints. Due to the fact that creation/destruction rates can be zero, the whole analysis of the long time behavior becomes quite delicate. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density remained open for most KCSM (with the exception of the East model in d=1 Aldous and P. Diaconis, J. Stat. Phys. 107(5–6):945–975 2002). Here we review a novel multi-scale approach which we have developed in Cancrini et al. (Probab. Theory Relat. Fields 140:459–504, 2008; Lecture Notes in Mathematics, vol. 1970, pp. 307–340, Springer, 2009) trough which we: (i) prove positivity of the spectral gap in the whole ergodic region for a wide class of KCSM on ℤ d , (ii) establish (sometimes optimal) bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iii) prove pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations. In particular (i) above establishes exponential decay of auto-correlation functions disproving the stretched exponential decay which had been conjecture for some KCSM and (ii) disproves some of the scalings which had been extrapolated from numerical simulations for the relaxation times (inverse of the spectral gap).

A new characterization of Talagrand’s transport-entropy inequalities and applications

We show that Talagrand’s transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand’s inequality is stable under bounded perturbations.

Characterization of Talagrand’s transport-entropy inequalities in metric spaces

We give a characterization of transport-entropy inequalities in metric spaces. As an application we deduce that such inequalities are stable under bounded perturbation (Holley-Stroock perturbation Lemma).

Universality in one-dimensional hierarchical coalescence processes

Motivated by several models introduced in the physics literature to study the nonequilibrium coarsening dynamics of one-dimensional systems, we consider a large class of “hierarchical coalescence processes” (HCP). An HCP consists of an infinite sequence of coalescence processes {ξ(n)(⋅)}n≥1: each process occurs in a different “epoch” (indexed by n) and evolves for an infinite time, while the evolution in subsequent epochs are linked in such a way that the initial distribution of ξ(n+1) coincides with the final distribution of ξ(n). Inside each epoch the process, described by a suitable simple point process representing the boundaries between adjacent intervals (domains), evolves as follows. Only intervals whose length belongs to a certain epoch-dependent finite range are active, that is, they can incorporate their left or right neighboring interval with quite general rates. Inactive intervals cannot incorporate their neighbors and can increase their length only if they are incorporated by active neighbors. The activity ranges are such that after a merging step the newly produced interval always becomes inactive for that epoch but active for some future epoch. Without making any mean-field assumption we show that: (i) if the initial distribution describes a renewal process, then such a property is preserved at all later times and all future epochs; (ii) the distribution of certain rescaled variables, for example, the domain length, has a well-defined and universal limiting behavior as n→∞ independent of the details of the process (merging rates, activity ranges, …). This last result explains the universality in the limiting behavior of several very different physical systems (e.g., the East model of glassy dynamics or the Paste-all model) which was observed in several simulations and analyzed in many physics papers. The main idea to obtain the asymptotic result is to first write down a recursive set of nonlinear identities for the Laplace transforms of the relevant quantities on different epochs and then to solve it by means of a transformation which in some sense linearizes the system.

Facilitated Spin Models: Recent and New Results

Facilitated or kinetically constrained spin models (KCSM) are a class of interacting particle systems reversible w.r.t. to a simple product measure. Each dynamical variable (spin) is re-sampled from its equilibrium distribution only if the surrounding configuration fulfills a simple local constraint which does not involve the chosen variable itself. Such simple models are quite popular in the glass community since they display some of the peculiar features of glassy dynamics, in particular they can undergo a dynamical arrest reminiscent of the liquid/glass transition. Due to the fact that the jumps rates of the Markov process can be zero, the whole analysis of the long time behavior becomes quite delicate and, until recently, KCSM have escaped a rigorous analysis with the notable exception of the East model. In these notes we will mainly review several recent mathematical results which, besides being applicable to a wide class of KCSM, have contributed to settle some debated questions arising in numerical simulations made by physicists. We will also provide some interesting new extensions. In particular we will show how to deal with interacting models reversible w.r.t. to a high temperature Gibbs measure and we will provide a detailed analysis of the so called one spin facilitated model on a general connected graph.

Sub-gaussian measures and associated semilinear problems.

We study the existence, smoothing properties and the long time behaviour for a class of nonlinear Cauchy problems in infinite dimensions under the assumption of F-Sobolev inequalities.

The logarithmic Sobolev constant of the lamplighter

We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application.