Hexagon

math.PR — Probability

On the escape rate of favorite sites of planar random walks

Contributed by Heng Ma

For planar simple random walk, the favorite sites at time $n$ are the sites whose local time at time $n$ is maximal. We prove that, almost surely, for every $\gamma>1/2$ and every $c>0$ , all favorite sites lie outside the ball centered at the origin with radius $c\sqrt n/(\log n)^\gamma$ for all sufficiently large $n$ . At the critical exponent $\gamma=1/2$ , almost surely, for every $c>0$ , the entire favorite sites lies within distance $c\sqrt{n/\log n}$ of the origin infinitely often.

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Oct 01, 2026PDF ↗View work →

hexagon:2609.00144v1

math.PR — Probability

On the Asymptotic $W_1$ Cost of Two-Dimensional Semi-Discrete Matching

Contributed by Heng Ma

Let $X_1,X_2,\ldots$ be independent uniform points in the unit square and let $\lambda$ denote Lebesgue probability measure. We prove that the normalized expected $W_1$ distance between the empirical measure and Lebesgue measure converges to a positive constant: \[ \lim_{N\to\infty} \sqrt{\frac{N}{\log N}}\, \mathbb E W_1\!\left(\frac1N\sum_{i=1}^N\delta_{X_i},\lambda\right) =c_\star. \] The constant is characterized by the convex viscosity equation \[ \partial_tU=\frac1{4\pi}\sqrt{\det D^2U}, \qquad U(0,q)=|q|. \] Its solution $U$ is unique under certain growth and regularity conditions, and the limiting constant $c_\star=U(1,0)/\sqrt2$ .

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Sep 30, 2026PDF ↗View work →

Works in math.PR

On the escape rate of favorite sites of planar random walks

On the Asymptotic W1W_1W1​ Cost of Two-Dimensional Semi-Discrete Matching

On the Asymptotic $W_1$ Cost of Two-Dimensional Semi-Discrete Matching