402 research outputs found
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
Incommensurability in the magnetic excitations of the bilinear-biquadratic spin-1 chain
We study the magnetic excitation spectrum of the S=1 quantum Heisenberg spin
chain with Hamiltonian : H = sum_i cos(theta) S_i S_i+1 + sin(theta) (S_i
S_i+1)^2. We focus on the range -pi/4 < theta < +pi/4 where the spin chain is
in the gapped Haldane phase. The excitation spectrum and static structure
factor is studied using direct Lanczos diagonalization of small systems and
density-matrix renormalization group techniques combined with the single-mode
approximation. The magnon dispersion has a minimum at q=pi until a critical
value theta_c = 0.38 is reached at which the curvature (velocity) vanishes.
Beyond this point, which is distinct from the VBS point and the Lifshitz point,
the minimum lies at an incommensurate value that goes smoothly to 2pi/3 when
theta approaches pi/4, the Lai-Sutherland point. The mode remains isolated from
the other states: there is no evidence of spinon deconfinement before the point
theta =+pi/4. These findings explain recent observation of the magnetization
curve M approx (H -H_c)^1/4 for theta =theta_c.Comment: 14 pages, 8 encapsulated figures, REVTeX 3.
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