Connected Operators for the Totally Asymmetric Exclusion Process

We fully elucidate the structure of the hierarchy of the connected operators that commute with the Markov matrix of the Totally Asymmetric Exclusion Process (TASEP). We prove for the connected operators a combinatorial formula that was conjectured in a previous work. Our derivation is purely algebraic and relies on the algebra generated by the local jump operators involved in the TASEP. Keywords: Non-Equilibrium Statistical Mechanics, ASEP, Exact Results, Algebraic Bethe Ansatz.Comment: 10 page

A Monte-Carlo study of meanders

We study the statistics of meanders, i.e. configurations of a road crossing a river through "n" bridges, and possibly winding around the source, as a toy model for compact folding of polymers. We introduce a Monte-Carlo method which allows us to simulate large meanders up to n = 400. By performing large "n" extrapolations, we give asymptotic estimates of the connectivity per bridge R = 3.5018(3), the configuration exponent gamma = 2.056(10), the winding exponent nu = 0.518(2) and other quantities describing the shape of meanders. Keywords : folding, meanders, Monte-Carlo, treeComment: 12 pages, revtex, 11 eps figure

Random incidence matrices: moments of the spectral density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices : any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semi-circle of "small" eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit), we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e=2.72... is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix. Keywords: random graphs, random matrices, sparse matrices, incidence matrices spectrum, momentsComment: 39 pages, 9 figures, Latex2e, [v2: ref. added, Sect. 4 modified

Core percolation in random graphs: a critical phenomena analysis

We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the "core". In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at "alpha = e = 2.718...": below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs. Key words: random graphs, leaf removal, core percolation, critical exponents, combinatorial optimization, finite size scaling, Monte-Carlo.Comment: 15 pages, 9 figures (color eps) [v2: published text with a new Title and addition of an appendix, a ref. and a fig.

Family of Commuting Operators for the Totally Asymmetric Exclusion Process

The algebraic structure underlying the totally asymmetric exclusion process is studied by using the Bethe Ansatz technique. From the properties of the algebra generated by the local jump operators, we explicitly construct the hierarchy of operators (called generalized hamiltonians) that commute with the Markov operator. The transfer matrix, which is the generating function of these operators, is shown to represent a discrete Markov process with long-range jumps. We give a general combinatorial formula for the connected hamiltonians obtained by taking the logarithm of the transfer matrix. This formula is proved using a symbolic calculation program for the first ten connected operators. Keywords: ASEP, Algebraic Bethe Ansatz. Pacs numbers: 02.30.Ik, 02.50.-r, 75.10.Pq.Comment: 26 pages, 1 figure; v2: published version with minor changes, revised title, 4 refs adde

Phases of random antiferromagnetic spin-1 chains

We formulate a real-space renormalization scheme that allows the study of the effects of bond randomness in the Heisenberg antiferromagnetic spin-1 chain. There are four types of bonds that appear during the renormalization flow. We implement numerically the decimation procedure. We give a detailed study of the probability distributions of all these bonds in the phases that occur when the strength of the disorder is varied. Approximate flow equations are obtained in the weak-disorder regime as well as in the strong disorder case where the physics is that of the random singlet phase.Comment: 29 pages, 12 encapsulated Postscript figures, REVTeX 3.

Spectral gap of the totally asymmetric exclusion process at arbitrary filling

We calculate the spectral gap of the Markov matrix of the totally asymmetric simple exclusion process (TASEP) on a ring of L sites with N particles. Our derivation is simple and self-contained and extends a previous calculation that was valid only for half-filling. We use a special property of the Bethe equations for TASEP to reformulate them as a one-body problem. Our method is closely related to the one used to derive exact large deviation functions of the TASEP