相关论文: Random matrix theory over finite fields: a survey

We develop the theory of copartitions, which are a generalization of partitions with connections to many classical topics in partition theory, including Rogers-Ramanujan partitions, theta functions, mock theta functions, partitions with…

This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric…

Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-d integrable model with the $1/r^2$ interaction (the Calogero-Sutherland-Moser system), and…

This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid…

Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically.…

Composed ensembles of random unitary matrices are defined via products of matrices, each pertaining to a given canonical circular ensemble of Dyson. We investigate statistical properties of spectra of some composed ensembles and demonstrate…

Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class…

We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not…

The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…

Many complex systems are characterized by intriguing spatio-temporal structures. Their mathematical description relies on the analysis of appropriate correlation functions. Functional integral techniques provide a unifying formalism that…

Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets being enumerated by…

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the…

Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational…

We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…

This article is a short review on the relationship between convergent matrix integrals, formal matrix integrals, and combinatorics of maps. We briefly summarize results developed over the last 30 years, as well as more recent discoveries.…

Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account on various martingale-based methods for over- and under-approximating…

In recent years,constructive field techniques and the method of renormalization group around extended singularities have been applied to the weak coupling regime of the Anderson Model. It has allowed to clarify the relationship between this…

We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as…

For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to a given pattern. For large approximating matrices, we observe that the…

This chapter collects several probabilistic tools that proved to be useful in the analysis of randomized search heuristics. This includes classic material like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics like…