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

Optimizing evacuation flow in a two-channel exclusion process

We use a basic setup of two coupled exclusion processes to model a stylised situation in evacuation dynamics, in which evacuees have to choose between two escape routes. The coupling between the two processes occurs through one common point at which particles are injected, the process can be controlled by directing incoming individuals into either of the two escape routes. Based on a mean-field approach we determine the phase behaviour of the model, and analytically compute optimal control strategies, maximising the total current through the system. Results are confirmed by numerical simulations. We also show that dynamic intervention, exploiting fluctuations about the mean-field stationary state, can lead to a further increase in total current.Comment: 16 pages, 6 figure

Supersymmetry on Jacobstahl lattices

It is shown that the construction of Yang and Fendley (2004 {\it J. Phys. A: Math.Gen. {\bf 37}} 8937) to obtainsupersymmetric systems, leads not to the open XXZ chain with anisotropy $\Delta =-{1/2}$ but to systems having dimensions given by Jacobstahl sequences.For each system the ground state is unique. The continuum limit of the spectra of the Jacobstahl systems coincide, up to degeneracies, with that of the $U_q(sl(2))$ invariant XXZ chain for $q=\exp(i\pi/3)$. The relation between the Jacobstahl systems and the open XXZ chain is explained.Comment: 6 pages, 0 figure

Bethe Ansatz solution of a decagonal rectangle triangle random tiling

A random tiling of rectangles and triangles displaying a decagonal phase is solved by Bethe Ansatz. Analogously to the solutions of the dodecagonal square triangle and the octagonal rectangle triangle tiling an exact expression for the maximum of the entropy is found.Comment: 17 pages, 4 figures, some remarks added and typos correcte

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

A refined Razumov-Stroganov conjecture II

We extend a previous conjecture [cond-mat/0407477] relating the Perron-Frobenius eigenvector of the monodromy matrix of the O(1) loop model to refined numbers of alternating sign matrices. By considering the O(1) loop model on a semi-infinite cylinder with dislocations, we obtain the generating function for alternating sign matrices with prescribed positions of 1's on their top and bottom rows. This seems to indicate a deep correspondence between observables in both models.Comment: 21 pages, 10 figures (3 in text), uses lanlmac, hyperbasics and epsf macro

Exact Ground State and Finite Size Scaling in a Supersymmetric Lattice Model

We study a model of strongly correlated fermions in one dimension with extended N=2 supersymmetry. The model is related to the spin $S=1/2$ XXZ Heisenberg chain at anisotropy $\Delta=-1/2$ with a real magnetic field on the boundary. We exploit the combinatorial properties of the ground state to determine its exact wave function on finite lattices with up to 30 sites. We compute several correlation functions of the fermionic and spin fields. We discuss the continuum limit by constructing lattice observables with well defined finite size scaling behavior. For the fermionic model with periodic boundary conditions we give the emptiness formation probability in closed form.Comment: 4 pages, 4 eps figure

Autocorrelations in the totally asymmetric simple exclusion process and Nagel-Schreckenberg model

We study via Monte Carlo simulation the dynamics of the Nagel-Schreckenberg model on a finite system of length L with open boundary conditions and parallel updates. We find numerically that in both the high and low density regimes the autocorrelation function of the system density behaves like 1-|t|/tau with a finite support [-tau,tau]. This is in contrast to the usual exponential decay typical of equilibrium systems. Furthermore, our results suggest that in fact tau=L/c, and in the special case of maximum velocity 1 (corresponding to the totally asymmetric simple exclusion process) we can identify the exact dependence of c on the input, output and hopping rates. We also emphasize that the parameter tau corresponds to the integrated autocorrelation time, which plays a fundamental role in quantifying the statistical errors in Monte Carlo simulations of these models.Comment: 7 pages, 6 figure

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