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Register a new userDistribution of the random walk conditioned on survival among quenched Bernoulli obstacles
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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.
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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 $a>0$.
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 random walk on dynamical percolation on the discrete torus $mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound.
Following similar analysis to that in Lacoin (PTRF 159, 777-808, 2014), we can show that the quenched critical point for self-avoiding walk on random conductors on the d-dimensional integer lattice is almost surely a constant, which does not depend on the location of the reference point. We provide its upper and lower bounds that are valid for all dimensions.
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