We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation @math is proportional to @math where @math is a positive parameter and @math is the number of inversions in @math . In our main result we show that when @math , the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in [Bhatnagar and Peled, PTRF, 2015]. This is in contrast to the case when @math where the limiting distribution of the LIS when scaled appropriately is the GUE Tracy-Widom distribution. We also obtain a law of large numbers for the length of the longest decreasing subsequence (LDS) and identify the precise constant in the order of the expectation, answering a further open question in [Bhatnagar and Peled, PTRF, 2015].
Recently, further progress been made in the analysis of the empirical measure of points corresponding to the Boltzmann-Gibbs measure of Mueller and Starr. Mukherjee @cite_11 determined the large deviation rate function of the empirical measure of points and Starr and Walters recently showed that the large deviation principle has a unique optimizer @cite_20 .
@cite_20 @cite_11
2063553966 2162928332 1999102639 2061282787
We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probability that the random graph is connected. For any finite colored graph we define the empirical neighborhood measure, which counts the number of vertices of a given color connected to a given number of vertices of each color, and the empirical pair measure, which counts the number of edges connecting each pair of colors. For a class of models of sparse colored random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distribution of Erdos-Renyi graphs near criticality. In this work we consider a problem related to the equilibrium Statistical Mechanics of Spin Glasses, namely the study of the Gibbs measure of the Random Energy Model. For solving this problem new results of independent interest on sums of spacings for i.i.d. Gaussian random variables are presented. Then we give a precise description of the support of the Gibbs measure below the critical temperature. Given a finite typed rooted tree T with n vertices, the empirical subtree measure is the uniform measure on the n typed subtrees of T formed by taking all descendants of a single vertex. We prove a large deviation principle in n, with explicit rate function, for the empirical subtree measures of multitype Galton–Watson trees conditioned to have exactly n vertices. In the process, we extend the notions of shift-invariance and specific relative entropy—as typically understood for Markov fields on deterministic graphs such as Zd—to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with n vertices.