Hitting probabilities and uniformly $S$-transient subgraphs

TL;DR

The paper investigates exit/hitting probabilities for subgraphs embedded in larger graphs under two-sided Gaussian heat-kernel (Harnack) assumptions. It develops both global and refined, geometry-aware bounds for hitting probabilities, first via a boundary-volume ratio bound $\psi_K(x) \le \sum_{n\ge d_x^2} C/W(x,\sqrt{n})$ and then, under inner uniformity, through harmonic profiles and $h$-transforms to obtain two-sided estimates. Central to the methodology are Neumann/Dirichlet kernels, harmonic profiles $h$, and the $h$-transform, which together connect hitting probabilities to transformed volumes $V_h$ and boundary geometry. The results yield both general upper bounds and two-sided bounds for harmonic measures, with extensive lattice examples illustrating $S$-transience versus uniform $S$-transience and drawing connections to Wiener's test. Overall, the work provides a framework for understanding how subgraph geometry and harmonic structure govern survival probabilities of random walks in glued or perforated graphs, with implications for gluing constructions and boundary-harmonic analysis.

Abstract

We study the probability that a random walk started inside a subgraph of a larger graph exits that subgraph (or, equivalently, hits the exterior boundary of the subgraph). Considering the chance a random walk started in the subgraph never leaves the subgraph leads to a notion we call "survival" transience, or $S$-transience. In the case where the heat kernel of the larger graph satisfies two-sided Gaussian estimates, we prove an upper bound on the probability of hitting the boundary of the subgraph. Under the additional hypothesis that the subgraph is inner uniform, we prove a two-sided estimate for this probability. The estimate depends upon a harmonic function in the subgraph. We also provide two-sided estimates for related probabilities, such as the harmonic measure (the chance the walk exits the subgraph at a particular point on its boundary).

Figures (3)

• Figure 1: Let $\widehat{\Gamma}$ be the full ten edges by ten edges square as on the left. Take $\Gamma$ to be $\widehat{\Gamma}$ minus the red points. The red points are $\partial \Gamma$, and the blue points are $\partial_I \Gamma$. Then $d(x, \partial \Gamma) = 4$ and $d(y, \partial \Gamma) = 3,$ and both of these distances are achieved by the same point in $\partial \Gamma,$ call it $z.$ Note $d_\Gamma(x,y) = 19 > d_{\Gamma}(x,z) + d_\Gamma(y,z) = 7.$ The correct way to think of this is duplicating the red points of $\partial \Gamma$ as shown in the right figure.
• Figure 2: The blue "flyswatter," which we imagine continues infinitely in both directions in a similar manner. Although this picture is in two dimensions, we think of this in a higher dimensional space. Note how there are black points in-between the blue points, and it is easy to see distance in $\mathbb{Z}^d$ would not be changed significantly by avoiding the blue points when $d \geq 4.$
• Figure 3: For $\alpha=1/2,$ we take the lattice points inside of the parabola $x_1=x_2^2$ as our set $K.$ This figure is the $x_1x_2$-plane that lives inside $\mathbb{Z}^4.$

Theorems & Definitions (80)

• Definition 1.1: Controlled Weights
• Definition 1.2: Doubling
• Definition 1.3: Poincaré Inequality
• Definition 1.4: Uniformly Lazy
• Definition 1.5: Harmonic Function
• Definition 1.6: Elliptic Harnack Inequality
• Definition 1.7: Solution of Discrete Heat Equation
• Definition 1.8: Parabolic Harnack Inequality
• Theorem 1.1: Theorem 1.7 in Delmotte_PHI
• Definition 1.9: Harnack Graph