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arXiv · 2610.14337

Variance Bounds for Minimum Perfect Matching Problem by Random Walk on Multinomial Distribution

Shi Feng
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Abstract

We establish variance upper bounds for the minimum perfect matching problem in general metric spaces under a geometric partition condition. As a principal application, let $H_n$ be the minimum perfect matching cost of $n$ independent uniform points in $[0,1]^d$, where $n$ is even and $d\ge2$. We prove $$\operatorname{Var}(H_n)\le C d n^{1-2/d},$$ where $C$ is independent of $n$ and $d$. This removes the logarithmic losses in earlier upper bounds in every dimension (Talagrand, 1995); in particular, the variance remains bounded in dimension two, answering an open question (Rhee, 1994). The proof relies on constructing a rapidly mixing Markov process with a multinomial stationary distribution.

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Shi Feng. 2026-11-03. Variance Bounds for Minimum Perfect Matching Problem by Random Walk on Multinomial Distribution. https://arxiv.org/abs/2610.14337

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