Semiclassical limits of quantum partition functions on infinite graphs

Abstract

We prove that if $H$ denotes the operator corresponding to the canonical Dirichlet form on a possibly locally infinite weighted graph $(X,b,m)$, and if $v:X\to \mathbb{R}$ is such that $H+v/\hbar$ is well-defined as a form sum for all $\hbar >0$, then the quantum partition function $\mathrm{tr}(\mathrm{e}^{-\beta \hbar ( H + v/\hbar)})$ satisfies $$ \mathrm{tr}(\mathrm{e}^{-\beta \hbar ( H + v/\hbar)})\xrightarrow[]{\hbar\to 0+}\sum_{x\in X} \mathrm{e}^{-\beta v(x)} \text{ for all $\beta>0$}, $$ regardless of the fact whether $\mathrm{e}^{-\beta v}$ is apriori summable or not. We also prove natural generalizations of this semiclassical limit to a large class of covariant Schr\"odinger operators that act on sections in Hermitian vector bundle over $(X,m,b)$, a result that particularly applies to magnetic Schr\"odinger operators that are defined on $(X,m,b)$

Note on uniformly transient graphs

“…Recently, however, there has been an outburst of all sorts of studies of Laplacians on graphs with general measures, see e.g. [2,6,7,11,14,15,16,17,20,22,25,27,28,33,32,37,38,40,45,46,47,57] and references therein. In some sense, a comparable development can be seen in the study of manifolds.…”

Recent probabilistic results on covariant Schrödinger operators on infinite weighted graphs

“…, and one often has C(b, m) < ∞ in applications (cf. [5] and the references therein for more details on these facts). Let us explain the analogy ofQ Φ,0 to the Riemannian case (2): Firstly, we have the edge vector bundle…”

“…Let us remark here that (X, b, m) is completely arbitrary in these results (in particular, (X, b) may be locally infinite, and we allow inf m = 0). Furthermore, if (X, b) does not support a symmetry which is respected by m and P • appropriately, then the Golden-Thompson bound (8) does not follow directly from (7), but rather from a combination of (7) for V = 0 with the abstract operator variant of the Golden-Thompson bound [6,5] (and a combination of geometric and functional analytic approximation arguments). The proof of (9) uses semigroup domination and the corresponding result in the scalar "nonmagnetic" situation, which itself makes full use of the path properties of X.…”

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

“…Note added: Let us mention the follow up papers by the first named author [14,15] which treat Feynman-Kac formulae and semiclassical limits for covariant Schrödinger semigroups on Hermitian vector bundles over infinite weighted graphs. These papers are heavily building on the results presented here.…”