Abstract. We consider the Potts model with q colors on a sequence of weighted graphs with adjacency matrices An, allowing for both positive and negative weights. Under a mild regularity condition on An we show that the mean-field prediction for the log partition function is asymptotically correct, whenever tr(A 2 n ) = o(n). In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to +∞. Using this, we establish the universality of the limiting log partition function of the ferromagnetic Potts model for a sequence of asymptotically regular graphs, and that of the Ising model for bi-regular bipartite graphs in both ferromagnetic and anti-ferromagnetic domain. We also derive a large deviation principle for the empirical measure of the colors for the Potts model on asymptotically regular graphs.

Abstract. In this paper we study Ollivier's coarse Ricci-curvature for graphs, and obtain exact formulas for the Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature which hold for a wide class of graphs. We also obtain a general lower bound on the Ricci-curvature involving the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize Ricci-flat graphs of girth 5, and give the first necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci-curvature of random bipartite graphs

Asymptotics of the normalizing constant is computed for a class of one parameter exponential families on permutations which includes Mallows model with Spearmans's Footrule and Spearman's Rank Correlation Statistic. The MLE, and a computable approximation of the MLE are shown to be consistent. The pseudo-likelihood estimator of Besag is shown to be $\sqrt{n}$-consistent. An iterative algorithm (IPFP) is proved to converge to the limiting normalizing constant. The Mallows model with Kendall's Tau is also analyzed to demonstrate flexibility of the tools of this paper.Comment: The presentation of the paper is changed. Proof of consistency of MLE and an approximation to the MLE is included. The convergence of the discrete IPFP algorithm is analyzed as opposed to the continuous versio

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (non-normal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erdős and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.2010 Mathematics Subject Classification. 05C15, 60C05, 60F05, 05D99.

Consider random polynomial n i=0 aix i of independent meanzero normal coefficients ai, whose variance is a regularly varying function (in i) of order α. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in [0, 1] with probability n −bα+o(1) , and no roots in (1, ∞) with probability n −b 0 +o(1) , hence for n even, it has no real roots with probability n −2bα −2b 0 +o(1) . Here, bα = 0 when α ≤ −1 and otherwise bα ∈ (0, ∞) is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution φ d (x, t) to the d-dimensional heat equation initiated by a Gaussian white noise φ d (x, 0), we confirm that the probability of φ d (x, t) = 0 for all t ∈ [1, T ], is T −bα+o(1) , for α = d/2 − 1.