Sparse random matrices: the eigenvalue spectrum revisited

We revisit the derivation of the density of states of sparse random matrices. We derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit. Using the iterative scheme introduced by Biroli and Monasson [J. Phys. A 32, L255 (1999)] we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration. The combination of the two methods yields a very simple simple geometric interpretation of the tails of the spectrum. We test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum.Comment: 18 pages, 7 figures. Accepted version, minor corrections, references adde

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work for the ASEP with parameters gamma=delta=0. Using our first result and also results of Uchiyama-Sasamoto-Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980's there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.Comment: An announcement of these results appeared here: http://www.pnas.org/content/early/2010/03/25/0909915107.abstract This version of the paper has updated references and corrects a gap in the proof of Proposition 6.11 which was in the published versio

Double Scaling and Finite Size Corrections in sl(2) Spin Chain

We find explicit expressions for two first finite size corrections to the distribution of Bethe roots, the asymptotics of energy and high conserved charges in the sl(2) quantum Heisenberg spin chain of length J in the thermodynamical limit J->\infty for low energies E\sim 1/J. This limit was recently studied in the context of integrability in perturbative N=4 super-Yang-Mills theory. We applied the double scaling technique to Baxter equation, similarly to the one used for large random matrices near the edge of the eigenvalue distribution. The positions of Bethe roots are described near the edge by zeros of Airy function. Our method can be generalized to any order in 1/J. It should also work for other quantum integrable models.Comment: 23 page

Multiple orthogonal polynomials, d-orthogonal polynomials, production matrices, and branched continued fractions

I analyze an unexpected connection between multiple orthogonal polynomials, d-orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot's combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal