The first order convergence law fails for random perfect graphs

We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on $n$ vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that $G_n$ satisfies it does not converge as $n\to\infty$.Comment: 11 pages. Minor corrections since last versio

The complexity of random ordered structures

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Around the circular law

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.Comment: Added: one reference and few comment

Probabilities of first-order sentences on sparse random relational structures: An application to definability on random CNF formulas

We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary s and a first-order theory T for s composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite s-structures that satisfy T and show that first-order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in s is linear. It is also shown that these limit probabilities are well behaved with respect to several parameters that represent the density of tuples in each relation R in the vocabulary s¿. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random k-CNF formula on n variables, where each possible clause occurs with probability ~c/nk-1¿, independently any first-order property of k-CNF formulas that implies unsatisfiability does almost surely not hold as n tends to infinity.Peer ReviewedPostprint (published version