# The Number of Open Paths in an Oriented ρ-Percolation Model

@article{Comets2007TheNO, title={The Number of Open Paths in an Oriented $\rho$-Percolation Model}, author={Francis Comets and Serguei Yu. Popov and M. Vachkovskaia}, journal={Journal of Statistical Physics}, year={2007}, volume={131}, pages={357-379} }

We study the asymptotic properties of the number of open paths of length n in an oriented ρ-percolation model. We show that this number is enα(ρ)(1+o(1)) as n→∞. The exponent α is deterministic, it can be expressed in terms of the free energy of a polymer model, and it can be explicitly computed in some range of the parameters. Moreover, in a restricted range of the parameters, we even show that the number of such paths is n−1/2Wenα(ρ)(1+o(1)) for some nondegenerate random variable W. We build…

## 14 Citations

### The number of open paths in oriented percolation

- Mathematics
- 2013

We study the number $N_n$ of open paths of length $n$ in supercritical oriented percolation on $\Zd \times \N$, with $d \ge 1$. We prove that on the percolation event $\{\inf N_n>0\}$, $N_n^{1/n}$…

### Directed polymer in random environment and last passage percolation

- Mathematics
- 2008

The sequence of random probability measures ν n that gives a path of length n , times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with…

### Limiting Results for the Free Energy of Directed Polymers in Random Environment with Unbounded Jumps

- Mathematics
- 2015

We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first…

### Minimal supporting subtrees for the free energy of polymers on disordered trees

- Mathematics
- 2008

We consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this model…

### Stochastic processes in random environment

- Mathematics
- 2009

We are interested in two probabilistic models of a process interacting with a random environment. Firstly, we consider the model of directed polymers in random environment. In this case, a polymer,…

### Lattice Versus Tree

- Physics
- 2017

In this chapter we deal with polymer models on different oriented graphs and compare them with the lattice case. As revealed by Derrida and Spohn in, many interesting questions can be answered on the…

### On the number of maximal paths in directed last-passage percolation

- MathematicsThe Annals of Probability
- 2020

We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically…

### Anisotropic bootstrap percolation in three dimensions

- MathematicsThe Annals of Probability
- 2020

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm…

## References

SHOWING 1-10 OF 27 REFERENCES

### Directed polymers in a random environment: path localization and strong disorder

- Mathematics
- 2003

We consider directed polymers in random environment. Under some mild assumptions on the environment, we prove here: (i) equivalence between the decay rate of the partition function and some natural…

### Asymptotic behavior of the critical probability for ρ-percolation in high dimensions

- Mathematics
- 2000

Abstract. We consider oriented bond or site percolation on ℤd+. In the case of bond percolation we denote by Pp the probability measure on configurations of open and closed bonds which makes all…

### Probabilistic analysis of directed polymers in a random environment: a review

- Materials Science
- 2004

Directed polymers in random environment can be thought of as a model of statistical mechanics in which paths of stochastic processes interact with a quenched disorder (impurities), depending on both…

### Directed polymers in random environment are diffusive at weak disorder

- Mathematics
- 2004

In this paper, we consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full…

### A remark concerning random walks with random potentials

- Mathematics
- 1995

We consider random walks where each path is equipped with a random weight which is stationary and independent in space and time. We show that under some assumptions the arising probability…

### Principles Of Random Walk

- Mathematics
- 1964

This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of…

### Majorizing multiplicative cascades for directed polymers in random media

- Mathematics
- 2005

In this note we give upper bounds for the free energy of discrete directed polymers in random media. The bounds are given by the so-called generalized multiplicative cascades from the statistical…

### ORNSTEIN–ZERNIKE THEORY FOR THE BERNOULLI BOND PERCOLATION ON Z d

- Mathematics, Physics
- 1999

We derive a precise Ornstein–Zernike asymptotic formula for the decay of the two-point function P p ( 0 ↔ x) of the Bernoulli bond percolation on the integer lattice Z d in any dimension d ≥ 2, in…

### Diffusion of directed polymers in a random environment

- Mathematics
- 1988

We consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time. In spatial dimensionsd>2, we establish that the behavior is…

### Ornstein-Zernike theory for the Bernoulli bond percolation on $\mathbb{Z}^d$

- Mathematics
- 2002

We derive a precise Ornstein-Zernike asymptotic formula for the decay of the two-point function P p (0 ↔ x) of the Bernoulli bond percolation on the integer lattice Z d in any dimension d ≥ 2, in any…