Fluctuations in random walks with random traps

A field theory for random walks on a lattice with random traps is introduced and related to the field theory for the Green's functions of an electron on a lattice with randomly placed repulsive potentials. The moments [${\phi }_N^k$] and [${\phi }_N^k\mathrm{}\left(0\right)\mathrm{}$] of the survival and return functions ${\phi }_N$ and ${\phi }_N\mathrm{}\left(0\right)\mathrm{}$ measuring, respectively, the probability that a random walk will survive or will return to the origin in $N$ steps for a given configuration of traps is calculated from instantons of the field theory. Both $\ln \left[{\phi }_N^k\right]$ and $\ln \left[{\phi }_N^k\mathrm{}\right(0\left)\right]$ are proportional to ${\mathrm{}\left(\mathrm{kN}\right)\mathrm{}}^{\frac{d}{\mathrm{}(d+2\mathrm{})\mathrm{}}}$ in $d$ dimensions with correction terms of order ${\mathrm{}\left(\mathrm{kN}\right)\mathrm{}}^{\frac{\mathrm{}(d-1\mathrm{})\mathrm{}}{\mathrm{}(d+2\mathrm{})\mathrm{}}}$. Thus [${\phi }_N^k$] is always much larger than ${\left[\phi N\right]}^k$ for large $N$ and similarly for [${\phi }_N\mathrm{}\left(0\right)\mathrm{}$].