Accurate estimation of the survival probability for trapping in two dimensions

In this work we study the mean survival probability $\Phi \mathrm{}(n,c)\mathrm{}$ of random walks on a two-dimensional lattice in the presence of traps of concentration c, as a function of the number of steps n. The computation of this quantity is performed indirectly by using the distribution of the number of sites visited $S_n.$ In order to achieve an accurate description of this distribution we use a combination of numerical techniques. The method allows an accurate calculation of $\Phi$ down to very small values (of the order of ${10}^{-100},$ for example), which is not possible via direct simulations. The survival probability is analyzed in terms of an asymptotic expansion, following the results of Donsker and Varadhan [Commun. Pure Appl. Math. 28, 525 (1975); 32, 721 (1979)], and by using the outcome of a scaling ansatz, as described in our earlier work.