149 research outputs found

### Semi-infinite TASEP with a Complex Boundary Mechanism

We consider a totally asymmetric exclusion process on the positive half-line.
When particles enter in the system according to a Poisson source, Liggett has
computed all the limit distributions when the initial distribution has an
asymptotic density. In this paper we consider systems for which particles enter
at the boundary according to a complex mechanism depending on the current
configuration in a finite neighborhood of the origin. For this kind of models,
we prove a strong law of large numbers for the number of particles entered in
the system at a given time. Our main tool is a new representation of the model
as a multi-type particle system with infinitely many particle types

### A Fredholm Determinant Representation in ASEP

In previous work the authors found integral formulas for probabilities in the
asymmetric simple exclusion process (ASEP) on the integer lattice. The dynamics
are uniquely determined once the initial state is specified. In this note we
restrict our attention to the case of step initial condition with particles at
the positive integers, and consider the distribution function for the m'th
particle from the left. In the previous work an infinite series of multiple
integrals was derived for this distribution. In this note we show that the
series can be summed to give a single integral whose integrand involves a
Fredholm determinant. We use this determinant representation to derive
(non-rigorously, at this writing) a scaling limit.Comment: 12 Pages. Version 3 includes a scaling conjectur

### Survival of contact processes on the hierarchical group

We consider contact processes on the hierarchical group, where sites infect
other sites at a rate depending on their hierarchical distance, and sites
become healthy with a constant recovery rate. If the infection rates decay too
fast as a function of the hierarchical distance, then we show that the critical
recovery rate is zero. On the other hand, we derive sufficient conditions on
the speed of decay of the infection rates for the process to exhibit a
nontrivial phase transition between extinction and survival. For our sufficient
conditions, we use a coupling argument that compares contact processes on the
hierarchical group with freedom two with contact processes on a renormalized
lattice. An interesting novelty in this renormalization argument is the use of
a result due to Rogers and Pitman on Markov functionals.Comment: Minor changes compared to previous version. Final version. 30 pages.
1 figur

### Formulas for ASEP with Two-Sided Bernoulli Initial Condition

For the asymmetric simple exclusion process on the integer lattice with
two-sided Bernoulli initial condition, we derive exact formulas for the
following quantities: (1) the probability that site x is occupied at time t;
(2) a correlation function, the probability that site 0 is occupied at time 0
and site x is occupied at time t; (3) the distribution function for the total
flux across 0 at time t and its exponential generating function.Comment: 18 page

### On a property of random-oriented percolation in a quadrant

Grimmett's random-orientation percolation is formulated as follows. The
square lattice is used to generate an oriented graph such that each edge is
oriented rightwards (resp. upwards) with probability $p$ and leftwards (resp.
downwards) otherwise. We consider a variation of Grimmett's model proposed by
Hegarty, in which edges are oriented away from the origin with probability $p$,
and towards it with probability $1-p$, which implies rotational instead of
translational symmetry. We show that both models could be considered as special
cases of random-oriented percolation in the NE-quadrant, provided that the
critical value for the latter is 1/2. As a corollary, we unconditionally obtain
a non-trivial lower bound for the critical value of Hegarty's
random-orientation model. The second part of the paper is devoted to higher
dimensions and we show that the Grimmett model percolates in any slab of height
at least 3 in $\mathbb{Z}^3$.Comment: The abstract has been updated, discussion has been added to the end
of the articl

### Multilayer parking with screening on a random tree

In this paper we present a multilayer particle deposition model on a random
tree. We derive the time dependent densities of the first and second layer
analytically and show that in all trees the limiting density of the first layer
exceeds the density in the second layer. We also provide a procedure to
calculate higher layer densities and prove that random trees have a higher
limiting density in the first layer than regular trees. Finally, we compare
densities between the first and second layer and between regular and random
trees.Comment: 15 pages, 2 figure

### Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes

We study the branching random walk on weighted graphs; site-breeding and
edge-breeding branching random walks on graphs are seen as particular cases. We
describe the strong critical value in terms of a geometrical parameter of the
graph. We characterize the weak critical value and relate it to another
geometrical parameter. We prove that, at the strong critical value, the process
dies out locally almost surely; while, at the weak critical value, global
survival and global extinction are both possible.Comment: 14 pages, corrected some typos and minor mistake

### Non equilibrium stationary state for the SEP with births and deaths

We consider the symmetric simple exclusion process in the interval
\La_N:=[-N,N]\cap\mathbb Z with births and deaths taking place respectively
on suitable boundary intervals $I_+$ and $I_-$, as introduced in De Masi et al.
(J. Stat. Phys. 2011). We study the stationary measure density profile in the
limit $N\to\infty

### Contact process in a wedge

We prove that the supercritical one-dimensional contact process survives in
certain wedge-like space-time regions, and that when it survives it couples
with the unrestricted contact process started from its upper invariant measure.
As an application we show that a type of weak coexistence is possible in the
nearest-neighbor ``grass-bushes-trees'' successional model introduced in
Durrett and Swindle (1991).Comment: 11 pages, 4 figure

### Hyperscaling in the Domany-Kinzel Cellular Automaton

An apparent violation of hyperscaling at the endpoint of the critical line in
the Domany-Kinzel stochastic cellular automaton finds an elementary resolution
upon noting that the order parameter is discontinuous at this point. We derive
a hyperscaling relation for such transitions and discuss applications to
related examples.Comment: 8 pages, latex, no figure

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