We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs $(G_n)_{n\geq0}$ converging locally to a Galton--Watson tree $T$ (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function $\phi_*$ of $T$. In the first part, we show that the adjacency operator associated with $T$ is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on $\phi_*$ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of $(G_n)_{n\geq 0}$. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.Comment: Published in at http://dx.doi.org/10.1214/10-AOP567 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We introduce a minimal theory of glass formation based on the ideas of molecular crowding and resultant string-like cooperative rearrangement, and address the effects of free interfaces. In the bulk case, we obtain a scaling expression for the number of particles taking part in cooperative strings, and we recover the Adam-Gibbs description of glassy dynamics. Then, by including thermal dilatation, the Vogel-Fulcher-Tammann relation is derived. Moreover, the random and string-like characters of the cooperative rearrangement allow us to predict a temperature-dependent expression for the cooperative length ξ of bulk relaxation. Finally, we explore the influence of sample boundaries when the system size becomes comparable to ξ. The theory is in agreement with measurements of the glass-transition temperature of thin polymer films, and allows quantification of the temperature-dependent thickness h m of the interfacial mobile layer.glass transition | cooperative rearrangement | thin films G lassy materials are ubiquitous in nature (1), and discussions about the glass transition involve many areas of physics, from molecular and spin glasses to hard-sphere jamming (2-7). Despite the intense interest in the dynamical slowing that accompanies glass formation, a single microscopic theory has yet to emerge (8-13). Nevertheless, the phenomenological approach of free volume (14) and the Doolittle ansatz (15) have been used to support the Vogel-Fulcher-Tammann (VFT) relation (16-18), which describes so many of the observed behaviors. Fundamental to glass formation are the suggestions that particles are increasingly crowded, and relaxation requires the cooperative participation of a growing number of particles. The hypothesis of a cooperatively rearranging region, as introduced by Adam and Gibbs (19), is appealing and has been observed in computational studies (20,21).The existence of a length scale ξ for cooperative rearrangement (22) has led to tremendous interest in confined glass formers, as initiated by ref. 23. Perhaps, the most active example of attempts to probe ξ is the study of glassy polymer films (24-26), where fascinating observations have been made. For the most studied case of polystyrene, reductions in the measured glass-transition temperature have been almost uniformly reported as the film thickness is reduced, both experimentally (27) and numerically (28). It has been further suggested that this apparent anomaly is linked to the observed existence of a more mobile interfacial layer (29-32). As a consequence, there have been many theoretical attempts to understand the thin-film glass transition, with varying degrees of complexity and success (33)(34)(35)(36)(37)(38).In this article, we present a simple analytical model for relaxation in glass-forming materials. First, from a microscopic molecular picture, the nature of the cooperative mechanism is explicitly defined and characterized as a function of density, and the Adam-Gibbs phenomenology is recovered. Then, by including thermal expansivity, we derive the VF...

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 Erdős-Rényi random graphs.