Pietro Caputo

Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph.

We consider the random reversible Markov kernel K obtained by assigning i.i.d. non negative weights to the edges of the complete graph over n vertices, and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an alpha-stable law, with alpha in (0,2). When 1<= \alpha <2, we show that for a suitable regularly varying sequence kappa_n of index 1-1/alpha, the limiting spectral distribution mu_alpha of kappa_n K coincides with the one of the random symmetric matrix of the un-normalized weights (Levy matrix with i.i.d. entries). In contrast, when 0< alpha <1, we show that the empirical spectral distribution of K converges without rescaling to a non trivial law wmu_alpha supported on [-1,1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of mu_alpha and wmu_alpha. We also study the limiting behavior of the invariant probability measure of K.

A note on wetting transition for gradient fields

We prove existence of a wetting transition for two classes of gradient fields which include: (1) The Continuous SOS model in any dimension and (2) The massless Gaussian model in dimension 2. Combined with a recent result proving the absence of such a transition for Gaussian models above 2 dimensions (Bolthausen et al., 2000.) J. Math. Phys. to appear), this shows in particular that absolute-value and quadratic interactions can give rise to completely different behavior.

Large deviations of empirical neighborhood distribution in sparse random graphs

Consider the Erdős–Renyi random graph on

Uniform Poincaré inequalities for unbounded conservative spin systems: the non-interacting case

n

Invariance principle for Mott variable range hopping and other walks on point processes

n vertices where each edge is present independently with probability

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

\lambda /n