Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeometry of the random walk range conditioned on survival among Bernoulli obstacles
107
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $mathbb{Z}^d$, $dgeq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $varrho_N=C N^{1/(d+2)}$ for any $dgeq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-epsilon)varrho_N$ for any $epsilon>0$, while the case $dgeq 3$ remains open. We complete the picture by showing that for any $dgeq 2$, the random walk range asymptotically contains a ball of radius $varrho_N-varrho_N^epsilon$ for some $epsilon in (0,1)$. Furthermore, we show that its boundary is of size at most $varrho_N^{d-1}(log varrho_N)^a$ for some $a>0$.
rate research
Read More
Place an obstacle with probability $1-p$ independently at each vertex of $mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time $n$ will be localized in a ball of volume asymptotically $dlog_{1/p}n$. In this work, we prove that this ball is free of obstacles, and we derive the limiting one-time distributions of the random walk conditioned on survival. Our proof is based on obstacle modifications and estimates on how such modifications affect the probability of the obstacle configurations as well as their associated Dirichlet eigenvalues, which is of independent interest.
We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d. random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general $d$-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the mixed probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice-versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of hitting times of Branching Random Walks in Random Environment (BRWRE).
For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M blocks such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size of each block is essentially linear in n. Let X_n be a random vector having the conditional distribution of X_n, conditioned on the total number of successes being at least k_n, where k_n is also essentially linear in n. Define Y_n similarly, but with success probabilities q_i>=p_i. We prove that the law of X_n converges weakly to a distribution that we can describe precisely. We then prove that sup Pr(X_n <= Y_n) converges to a constant, where the supremum is taken over all possible couplings of X_n and Y_n. This constant is expressed explicitly in terms of the parameters of the system.
We consider a random walk with a negative drift and with a jump distribution which under Cramers change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally-positive Levy %-Khinchin process conditioned not to overshoot level one.
We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in 86 for the volume of the range in dimension two.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
