Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model

Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

“…For i.i.d. conductances (1.1) even decides between a completely homogenizing phase, which we cover in this paper, and a completely localizing phase of the principal Dirichlet eigenvector, which was studied in [Fle16]. We thus extend the results of Faggionato [Fag12] and Boivin and Depauw [BD03].…”

“…If q < q c , then it is possible (and in the i.i.d. case even almost sure [Fle16]) that trapping structures as in Figure 3 appear, which immediately contradict a uniform Poincaré inequality.…”

Eigenvector localization in the heavy-tailed random conductance model

“…Note that the only reason why we have not generalized our findings to the first k eigenvectors in [Fle16], is that in [Fle16,Lemma 5.6] we rely on the property that the principal Dirichlet eigenvector does not change its sign, according to the Perron-Frobenius theorem. This is no longer true for the higher order eigenvectors.…”

“…This means that we are interested in the Dirichlet eigenfunctions and eigenvalues of the operator −L w in the box B n with zero Dirichlet conditions. In the recent paper [Fle16], we have shown that if γ := sup{q ≥ 0 : E[w −q ] < ∞} < 1/4 and certain regularity assumptions apply, then the principal Dirichlet eigenvector ψ (n) 1 of Problem (1.2) concentrates in a single site as n tends to infinity. To be more precise, let π z = x : x∼z w xz be the local speed measure, i.e., the inverse mean waiting time of the random walk generated by L w .…”

“…6]. To keep the present paper as short as possible, we refer the reader to our first article [Fle16] for more heuristics and references. However, we kept the present paper mostly self-contained.…”