Ian Meng Si | Searcharxiv

arXiv2026

Strategic geometry of competing first-passage random walks

Two competitors choose starting vertices for independent, constant-speed random walks, and each site is acquired by its first visitor. We study the spatial geometry of the resulting first-passage location game. On every finite path the optimal strategies are exactly the distributions supported on the central vertices. The proof combines reflecting-boundary harmonic barriers with a parameter-uniform aggregate estimate for product-chain exit probabilities. After diffusive rescaling, the complete two-start payoff landscape converges uniformly to the game between independent reflected Brownian motions. Its unique equilibrium concentrates at the midpoint, with explicit cubic stability. Beyond paths, attaching two leaves to every vertex of a clique of order k produces a 3k-vertex graph on which every exact optimal strategy randomizes over all k clique vertices; for k=2, this is a six-vertex tree with no pure equilibrium. The continuum best response to an endpoint is uniquely determined. A first-passage random-ranking representation relates the finite game to maximal lotteries without identifying it with nonstrategic painting, deterministic Voronoi allocation, or absorbing-trap placement. Parameter-uniform statements follow from analytic arguments or symbolic polynomial identities; identified finite exceptions and numerical enclosures have reproducible certificates.

math.PR↗

arXiv2026

Nonlocality of Cover-Time Changes Under Edge Addition

Let $G$ be a finite connected simple graph, let $uv$ be a nonedge, and let $s$ be a starting vertex. We study the exact change in the expected cover time of simple random walk when $uv$ is inserted. A killed Green matrix update, combined with target-set inclusion-exclusion, gives an exact formula using only the original graph. The response can have either sign. Our main result is a nonlocality theorem. For every radius $r\geq 1$, we construct two marked configurations whose ambient-degree-labelled radius-$r$ neighbourhoods at $s,u,v$ are isomorphic but whose fixed-start cover-time responses have opposite signs. The construction also matches the three marked degrees, the marked distance, and the effective resistance $R_{uv}$. The two graphs have different orders; equal-order nonlocality remains open. We complement this result with a conductance interpolation theorem and an occupation interpretation of its high-conductance coefficient. After contracting $u$ and $v$, that coefficient is a positive multiple of the expected pre-cover occupation of the contracted vertex, and its zero case is classified exactly. A path with two pendant insertion endpoints is solved for all starting vertices and shows a sharp change of sign across the start set.

math.PR↗