97 research outputs found
Full Current Statistics for a Disordered Open Exclusion Process
We consider the nonabelian sandpile model defined on directed trees by Ayyer,
Schilling, Steinberg and Thi\'ery (Commun. Math. Phys, 2013) and restrict it to
the special case of a one-dimensional lattice of sites which has open
boundaries and disordered hopping rates. We focus on the joint distribution of
the integrated currents across each bond simultaneously, and calculate its
cumulant generating function exactly. Surprisingly, the process conditioned on
seeing specified currents across each bond turns out to be a renormalised
version of the same process. We also remark on a duality property of the large
deviation function. Lastly, all eigenvalues and both Perron eigenvectors of the
tilted generator are determined.Comment: 14 pages, minor clarification
A finite variant of the Toom Model
We present results for a finite variant of the one-dimensional Toom model
with closed boundaries. We show that the steady state distribution is not of
product form, but is nonetheless simple. In particular, we give explicit
formulas for the densities and some nearest neighbour correlation functions. We
also give exact results for eigenvalues and multiplicities of the transition
matrix using the theory of -trivial monoids in joint work with A.
Schilling, B. Steinberg and N. M. Thi\'ery.Comment: Journal of Physics: Conference Series stylefile, 8 pages, 1 figure;
minor changes and clarification
A Statistical Model of Current Loops and Magnetic Monopoles
We formulate a natural model of current loops and magnetic monopoles for
arbitrary planar graphs, which we call the monopole-dimer model, and express
the partition function of this model as a determinant. We then extend the
method of Kasteleyn and Temperley-Fisher to calculate the partition function
exactly in the case of rectangular grids. This partition function turns out to
be a square of the partition function of an emergent monomer-dimer model when
the grid sizes are even. We use this formula to calculate the local monopole
density, free energy and entropy exactly. Our technique is a novel
determinantal formula for the partition function of a model of vertices and
loops for arbitrary graphs.Comment: 17 pages, 5 figures, significant stylistic revisions. In particular,
rewritten with a mathematical audience in mind. Numerous errors fixed. This
is the final published version. Maple program file can be downloaded from the
link on the right of this pag
Exact results for an asymmetric annihilation process with open boundaries
We consider a nonequilibrium reaction-diffusion model on a finite one
dimensional lattice with bulk and boundary dynamics inspired by Glauber
dynamics of the Ising model. We show that the model has a rich algebraic
structure that we use to calculate its properties. In particular, we show that
the Markov dynamics for a system of a given size can be embedded in the
dynamics of systems of higher sizes. This remark leads us to devise a technique
we call the transfer matrix Ansatz that allows us to determine the steady state
distribution and correlation functions. Furthermore, we show that the disorder
variables satisfy very simple properties and we give a conjecture for the
characteristic polynomial of Markov matrices. Lastly, we compare the transfer
matrix Ansatz used here with the matrix product representation of the steady
state of one-dimensional stochastic models.Comment: 18 page
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