Weak universality of dynamical Φ34\Phi^4_3: non-Gaussian noise

We consider a class of continuous phase coexistence models in three spatial dimensions. The fluctuations are driven by symmetric stationary random fields with sufficient integrability and mixing conditions, but not necessarily Gaussian. We show that, in the weakly nonlinear regime, if the external potential is a symmetric polynomial and a certain average of it exhibits pitchfork bifurcation, then these models all rescale to $\Phi^4_3$ near their critical point.Comment: 37 pages; updated introduction and reference

Matchings on infinite graphs

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page

Percolation on dense graph sequences

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained by keeping each edge independently with probability $p_n$. We show that the appearance of a giant component in $G_n(p_n)$ has a sharp threshold at $p_n=1/\lambda_n$. In fact, we prove much more: if $(G_n)$ converges to an irreducible limit, then the density of the largest component of $G_n(c/n)$ tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollob\'as, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.Comment: Published in at http://dx.doi.org/10.1214/09-AOP478 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org