Survival probability of a random walk among a Poisson system of moving traps.

*(English)*Zbl 1267.60109
Deuschel, Jean-Dominique (ed.) et al., Probability in complex physical systems. In honour of Erwin Bolthausen and Jürgen Gärtner. Selected papers based on the presentations at the two 2010 workshops. Berlin: Springer (ISBN 978-3-642-23810-9/hbk; 978-3-642-23811-6/ebook). Springer Proceedings in Mathematics 11, 119-158 (2012).

Let \(\{X:=X(t); t \geq 0\}\) be a simple symmetric random walk on \(\mathbb{Z}^d\) with jump rate \(\kappa>0\), and let \(\{Y_j^y; 0\leq j \leq N_y, y \in \mathbb{Z}^d\}\) be a collection of independent simple symmetric random walks on \(\mathbb{Z}^d\) with jump rate \(\rho>0\), where \(N_y\) is a number of walks that start at each \(y\in \mathbb{Z}^d\) at time 0. Assume that the \(N_y\), \(y\in \mathbb{Z}^d\), are i.i.d. Poisson distributed with mean \(\nu>0\) and \(\{Y_j^y:=Y_j^y(t), t \geq 0\}\) denotes the \(j\)-th walk starting at \(y\) at time 0. Let \(\xi(t,x)\) denote the number of walks \(Y\) at position \(x \in \mathbb{Z}^d\) at time \(t\). Walks \(Y\) are considered as traps, and at each time \(t\), the walk \(X\) is killed with the rate \(\gamma \xi(t,x)\), \(\gamma>0\). A quenched survival probability by time \(t\) is given by
\[
Z^\gamma_{t}:=\operatorname{E}^X \left[\exp\left\{\int_0^t\xi(s,X(s))\,ds\right\}\right]
\]
and annealed survival probability by time \(t\) is given by \(\operatorname{E}^\xi[Z^\gamma_{t}]\).

It is shown that the annealed survival probability decays asymptotically as \(e^{-\lambda_1 t^{1/2}}\) for \(d=1\), as \(e^{-\lambda_2 t/\log t}\) for \(d=2\), and as \(e^{-\lambda_d t}\) for \(d \geq 3\) where \(\lambda_1\) and \(\lambda_2\) can be identified explicitly. The quenched survival probability decays asymptotically as \(e^{-\overline{\lambda}_d t}\) for all \(d \geq 1\). A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous-time symmetric random walk increases under perturbation by a deterministic path.

The problem can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential.

For the entire collection see [Zbl 1235.60005].

It is shown that the annealed survival probability decays asymptotically as \(e^{-\lambda_1 t^{1/2}}\) for \(d=1\), as \(e^{-\lambda_2 t/\log t}\) for \(d=2\), and as \(e^{-\lambda_d t}\) for \(d \geq 3\) where \(\lambda_1\) and \(\lambda_2\) can be identified explicitly. The quenched survival probability decays asymptotically as \(e^{-\overline{\lambda}_d t}\) for all \(d \geq 1\). A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous-time symmetric random walk increases under perturbation by a deterministic path.

The problem can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential.

For the entire collection see [Zbl 1235.60005].

Reviewer: Alex V. Kolnogorov (Novgorod)

##### MSC:

60K37 | Processes in random environments |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82C22 | Interacting particle systems in time-dependent statistical mechanics |