1,319 research outputs found
Finite-size left-passage probability in percolation
We obtain an exact finite-size expression for the probability that a
percolation hull will touch the boundary, on a strip of finite width. Our
calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and
the results are expressed in terms of symplectic characters. In the large size
limit, we recover the scaling behaviour predicted by Schramm's left-passage
formula. We also derive a general relation between the left-passage probability
in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the
open XXZ chain with diagonal, complex boundary terms.Comment: 21 pages, 8 figure
Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon
The raise and peel model of a one-dimensional fluctuating interface (model A)
is extended by considering one source (model B) or two sources (model C) at the
boundaries. The Hamiltonians describing the three processes have, in the
thermodynamic limit, spectra given by conformal field theory. The probability
of the different configurations in the stationary states of the three models
are not only related but have interesting combinatorial properties. We show
that by extending Pascal's triangle (which gives solutions to linear relations
in terms of integer numbers), to an hexagon, one obtains integer solutions of
bilinear relations. These solutions give not only the weights of the various
configurations in the three models but also give an insight to the connections
between the probability distributions in the stationary states of the three
models. Interestingly enough, Pascal's hexagon also gives solutions to a
Hirota's difference equation.Comment: 33 pages, an abstract and an introduction are rewritten, few
references are adde
Exactly solvable quantum spin ladders associated with the orthogonal and symplectic Lie algebras
We extend the results of spin ladder models associated with the Lie algebras
to the case of the orthogonal and symplectic algebras $o(2^n),\
sp(2^n)$ where n is the number of legs for the system. Two classes of models
are found whose symmetry, either orthogonal or symplectic, has an explicit n
dependence. Integrability of these models is shown for an arbitrary coupling of
XX type rung interactions and applied magnetic field term.Comment: 7 pages, Late
Relaxation rate of the reverse biased asymmetric exclusion process
We compute the exact relaxation rate of the partially asymmetric exclusion
process with open boundaries, with boundary rates opposing the preferred
direction of flow in the bulk. This reverse bias introduces a length scale in
the system, at which we find a crossover between exponential and algebraic
relaxation on the coexistence line. Our results follow from a careful analysis
of the Bethe ansatz root structure.Comment: 22 pages, 12 figure
Partition function of the trigonometric SOS model with reflecting end
We compute the partition function of the trigonometric SOS model with one
reflecting end and domain wall type boundary conditions. We show that in this
case, instead of a sum of determinants obtained by Rosengren for the SOS model
on a square lattice without reflection, the partition function can be
represented as a single Izergin determinant. This result is crucial for the
study of the Bethe vectors of the spin chains with non-diagonal boundary terms.Comment: 13 pages, improved versio
Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
The asymmetric simple exclusion process with open boundaries, which is a very
simple model of out-of-equilibrium statistical physics, is known to be
integrable. In particular, its spectrum can be described in terms of Bethe
roots. The large deviation function of the current can be obtained as well by
diagonalizing a modified transition matrix, that is still integrable: the
spectrum of this new matrix can be also described in terms of Bethe roots for
special values of the parameters. However, due to the algebraic framework used
to write the Bethe equations in the previous works, the nature of the
excitations and the full structure of the eigenvectors were still unknown. This
paper explains why the eigenvectors of the modified transition matrix are
physically relevant, gives an explicit expression for the eigenvectors and
applies it to the study of atypical currents. It also shows how the coordinate
Bethe Ansatz developped for the excitations leads to a simple derivation of the
Bethe equations and of the validity conditions of this Ansatz. All the results
obtained by de Gier and Essler are recovered and the approach gives a physical
interpretation of the exceptional points The overlap of this approach with
other tools such as the matrix Ansatz is also discussed. The method that is
presented here may be not specific to the asymmetric exclusion process and may
be applied to other models with open boundaries to find similar exceptional
points.Comment: references added, one new subsection and corrected typo
Refined Razumov-Stroganov conjectures for open boundaries
Recently it has been conjectured that the ground-state of a Markovian
Hamiltonian, with one boundary operator, acting in a link pattern space is
related to vertically and horizontally symmetric alternating-sign matrices
(equivalently fully-packed loop configurations (FPL) on a grid with special
boundaries).We extend this conjecture by introducing an arbitrary boundary
parameter. We show that the parameter dependent ground state is related to
refined vertically symmetric alternating-sign matrices i.e. with prescribed
configurations (respectively, prescribed FPL configurations) in the next to
central row.
We also conjecture a relation between the ground-state of a Markovian
Hamiltonian with two boundary operators and arbitrary coefficients and some
doubly refined (dependence on two parameters) FPL configurations. Our
conjectures might be useful in the study of ground-states of the O(1) and XXZ
models, as well as the stationary states of Raise and Peel models.Comment: 11 pages LaTeX, 8 postscript figure
Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries
We analyze the Bethe ansatz equations describing the complete spectrum of the
transition matrix of the partially asymmetric exclusion process on a finite
lattice and with the most general open boundary conditions. We extend results
obtained recently for totally asymmetric diffusion [J. de Gier and F.H.L.
Essler, J. Stat. Mech. P12011 (2006)] to the case of partial symmetry. We
determine the finite-size scaling of the spectral gap, which characterizes the
approach to stationarity at large times, in the low and high density regimes
and on the coexistence line. We observe boundary induced crossovers and discuss
possible interpretations of our results in terms of effective domain wall
theories.Comment: 30 pages, 9 figures, typeset for pdflatex; revised versio
Structure of the two-boundary XXZ model with non-diagonal boundary terms
We study the integrable XXZ model with general non-diagonal boundary terms at
both ends. The Hamiltonian is considered in terms of a two boundary extension
of the Temperley-Lieb algebra.
We use a basis that diagonalizes a conserved charge in the one-boundary case.
The action of the second boundary generator on this space is computed. For the
L-site chain and generic values of the parameters we have an irreducible space
of dimension 2^L. However at certain critical points there exists a smaller
irreducible subspace that is invariant under the action of all the bulk and
boundary generators. These are precisely the points at which Bethe Ansatz
equations have been formulated. We compute the dimension of the invariant
subspace at each critical point and show that it agrees with the splitting of
eigenvalues, found numerically, between the two Bethe Ansatz equations.Comment: 9 pages Latex. Minor correction
Exact density profiles for fully asymmetric exclusion process with discrete-time dynamics
Exact density profiles in the steady state of the one-dimensional fully
asymmetric simple exclusion process on semi-infinite chains are obtained in the
case of forward-ordered sequential dynamics by taking the thermodynamic limit
in our recent exact results for a finite chain with open boundaries. The
corresponding results for sublattice parallel dynamics follow from the
relationship obtained by Rajewsky and Schreckenberg [Physica A 245, 139 (1997)]
and for parallel dynamics from the mapping found by Evans, Rajewsky and Speer
[J. Stat. Phys. 95, 45 (1999)]. By comparing the asymptotic results appropriate
for parallel update with those published in the latter paper, we correct some
technical errors in the final results given there.Comment: About 10 pages and 3 figures, new references are added and a
comparison is made with the results by de Gier and Nienhuis [Phys. Rev. E 59,
4899(1999)
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