The Annealed Parabolic Anderson Model on a Regular Tree

#### D. Wang, F. den Hollander

2024, v.30, Issue 1, 105-147

ABSTRACT

We study the total mass of the solution to the parabolic Anderson

model on a regular tree with an i.i.d. random potential whose marginal dis-

tribution is double-exponential. In earlier work we identified two terms in the

asymptotic expansion for large time of the total mass under the quenched law,

i.e., conditional on the realisation of the random potential. In the present paper

we do the same for the annealed law, i.e., averaged over the random potential.

It turns out that the annealed expansion differs from the quenched expansion.

The derivation of the annealed expansion is based on a new approach to control

the local times of the random walk appearing in the Feynman-Kac formula for

the total mass. In particular, we condition on the backbone to infinity of the

random walk, truncate and periodise the infinite tree relative to the backbone

to obtain a random walk on a finite subtree with a specific boundary condition,

employ the large deviation principle for the empirical distribution of Markov

renewal processes on finite graphs, and afterwards let the truncation level tend

to infinity to obtain an asymptotically sharp asymptotic expansion.

doi:10.61102/1024-2953-mprf.2024.30.1.005

Keywords: Parabolic Anderson model, Feynman-Kac formula, regular tree, doubleexponential random potential, backbone of random walk, annealed Lyapunov exponent, variational formula

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