CS45: Lévy processes and random walks in random and deterministic environments and their spectral theory

date: 7/15/2025, time: 16:00-17:30, room: ICS 25

Organizer: Kamil Kaleta & Katarzyna Pietruska-Pałuba (Wrocław University of Science and Technology & University of Warsaw)

Chair: Kamil Kaleta & Katarzyna Pietruska-Pałuba (Wrocław University of Science and Technology & University of Warsaw)

Upper heat kernel bounds for random walks on graphs with unbounded geometry

Christian Rose (University of Potsdam)

We consider heat kernels of continuous-time nearest neighbor random walks on discrete graphs. While it is known that Varadhan’s small-time asymptotics fails for such heat kernels, it is expected that for large times they are comparable to heat kernels of Riemannian manifolds. This talk presents a recent characterization of sharp Gaussian upper bounds in terms of volume doubling and Faber-Krahn inequalities on graphs with possibly unbounded geometry \([4,5]\). It constitutes the graph analogue of a celebrated result by Grigor’yan on Riemannian manifolds \([2]\). A non-sharp version for graphs with very bounded geometry can be found in \([1]\). Sharpness refers to the comparability of the heat kernel on the integers \([3]\). The implications of the characterization for unbounded geometry come with new information: heat kernel bounds are corrected by the vertex degree of the space-variables, and the corrections become small exponentially fast in time. The Faber-Krahn dimension in balls can be related to the the vertex degree growth and the doubling dimension.

Bibliography

\([1]\) M. T. Barlow. Random Walks and Heat Kernels on Graphs. London Mathematical Society Lecture Note Series. Cambridge University Press, 2017.

\([2]\) A. Grigor’yan. Heat Kernel and Analysis on Manifolds. AMS/IP studies in advanced mathematics. American Mathematical Society, 2009.

\([3]\) M. M. H. Pang. Heat kernels of Graphs. J. London Math. Soc., 47:50–64, 1993.

\([4]\) C. Rose. Off-diagonal upper heat kernel bounds on graphs with unbounded geometry. 2025. arXiv:2502.20239

\([5]\) C. Rose. Gaussian upper heat kernel bounds and Faber-Krahn inequalities on graphs. 2024. arXiv:2410.11715

Random walks and branching processes in a sparse random environment

Alicja Kołodziejska (JLU Gießen & University of Wroclaw)

I will briefly present the connection between transient nearest-neighbour random walks in random environment on \(\mathbb{Z}\) and certain Galton-Watson processes in random environment. This connection can be used to prove annealed limit theorems for first passage times and maximal local times of the random walk by examining the growth of the associated branching process. I will focus on the model known as a random walk in a sparse random environment, in which the walker moves symmetrically apart from some randomly chosen sites where random drift is imposed. This model may be thought of as an interpolation between the simple symmetric random walk and a random walk in i.i.d. random environment, in which random drifts are put at all sites. We will see how the interplay between the drift and the sparsity of the environment affects the asymptotic properties of the walk.

Bibliography

\([1]\) A. Kołodziejska, On favourite sites of a random walk in moderately sparse random environment, arXiv 2407.01206.

\([2]\) D. Buraczewski, P. Dyszewski, A. Iksanov, A. Marynych, and A. Roitershtein, Random walks in a moderately sparse random environment, Electronic Journal of Probability 24 (2019).

\([3]\) H. Kesten, M. Kozlov, and F. Spitzer, A limit law for random walk in a random environment, Compositio Mathematica 30 (1975).

\([4]\) A. Matzavinos, A. Roitershtein, and Y. Seol, Random walks in a sparserandom environment, Electronic Journal of Probability 21 (2016).

Lifshitz singularity for random Levy-Schroedinger operators with long range interactions

Katarzyna Pietruska-Pałuba (University of Warsaw)

In \([2]\) we have addressed the asymptotic behaviour of the integrated density of states (IDS) for random Lévy-Schrödinger operators \[H^\omega=\Phi(-\Delta)+V^\omega,\] where \(V^\omega\) is an alloy-type potential \[V^\omega(x)=\sum_{\mathbf i\in\mathbb R^d} q_{\mathbf i}(\omega) W(x-\mathbf i),\;\; x\in\mathbb R^d.\] Here \(q_{\mathbf i}\) are i.i.d. random variables and \(W\) is the potential profile. Our previous results \([1]\) were concerned with compactly supported profiles, and in present work we consider profiles of unbounded support. When \(W(|x|)\leq |x|^{-(d+\alpha)}\) for large \(|x|\) (\(\alpha\) depends on \(\Phi),\) then this behaviour is similar to that when \(W\) was of compact support. However, if this is not the case, the behaviour can depend on properties of random variables \(q_{\mathbf i}.\) In any case, Lifschitz singularity is present.

Both our previous and current results extend - to the case of nonlocal kinetic terms \(\Phi(-\Delta)\) - the results for the Laplacian of Kirsch and Simon from \([3]\).

Bibliography

\([1]\) Kaleta, K., Pietruska-Pałuba, K. (2025). "Lifshitz tail for long-range alloy-type models with Lévy operators", preprint.

\([2]\) Kaleta, K., Pietruska-Pałuba, K. "Lifschitz tail for continuous Anderson models driven by Lévy operators." Comm. Contemp. Math. 2020, 2050065 (46 pages).

\([3]\) Kirsch, W., Simon, B., "Lifshitz tails for periodic plus random potentials." J. Stat. Phys. vol. 42, no. 5/6, 1986, pp. 799-808.