We consider an electric network on the $d$-dimensional integer lattice with an edge between every two points $x$ and $y$. The conductance of the edge $\{x,y\}$ equals $\|x-y\|^{-s}$, for some $s>d$. We show that the random walk on this network is recurrent if and only if $d \in \{1,2\}$ and $s\geq 2d$. We also discuss how this result relates to the return properties of random walks on percolation clusters, particularly on the two-dimensional weight-dependent random connection model.

The theory of Abstract Wiener Spaces is the basis for many fundamental results of Gaussian measure theory: Large Deviations, Cameron-Martin theorems, Malliavin Calculus, Support theorems, etc. Analogues of these classical theorems exist also in the context of Gaussian Rough Paths and Regularity Structures. It is our goal to investigate the role of an “enhanced” Cameron-Martin subspace in this setting. In particular, we present two approaches to a generalization based on Large Deviation theory and apply them to examples of Rough Path theory and Regularity Structures.

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