arXiv subjects

Daoyi Wang

Publications and source records attributed to Daoyi Wang.

3 recordsLinked to original sources

arXiv2023

The parabolic Anderson model on a Galton-Watson tree with normalised Laplacian

In earlier work by den Hollander, König, and dos Santos, the asymptotics of the total mass of the solution to the parabolic Anderson model was studied on an almost surely infinite Galton-Watson tree with an i.i.d. potential having a double-exponential distribution. The second-order contribution to this asymptotics was identified in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The present paper extends this work to the degree-normalised Laplacian. The normalisation causes the Laplacian to be non-symmetric and which leads to different spectral properties. We find that the leading order asymptotics of the total mass remains the same, while the second-order correction coming from the variational formula is different. We also find that the optimiser of the variational formula is again an infinite tree with minimal degrees. Both of these results are shown to hold under much milder conditions than for the regular Laplacian.

math.PR↗

arXiv2023

The annealed parabolic Anderson model on a regular tree

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 distribution 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.

math.PR↗

arXiv2021

The Parabolic Anderson Model on a Galton-Watson tree revisited

In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is double-exponential. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. The second term contains a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. The present paper extends the analysis to degree distributions with unbounded support. We identify the weakest condition on the tail of the degree distribution under which the arguments in [1] can be pushed through. To do so we need to control the occurrence of large degrees uniformly in large subtrees of the Galton-Watson tree.

math.PR↗