AI 中文总结
本文研究d≥2维径向一次激发随机游走，证明其常返性及n步轨迹渐近为欧氏球的球形形状定理，证实了Kozma 2007年的相关猜想。
AI 中文摘要
我们考虑d≥2维整数格点Z^d上的一次激发随机游走：该游走首次访问某点时，会以恒定强度β>0偏向原点，后续再次访问该点时则如同简单对称随机游走般移动。我们证明该游走是常返的，并建立了几乎必然的球形形状定理：n步后的轨迹渐近为以原点为中心的欧氏球，半径与n^(1/(d+1))成正比，且局部时具有渐近锥形分布。该结果证实了Kozma（2007）针对此模型轨迹形状提出的猜想。
英文摘要
We consider a once-excited random walk on $\mathbb{Z}^d$, $d \ge 2$, where the walk on its first visit has a bias of constant strength $β>0$ toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape theorem: the trace after $n$ steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to $n^{1/(d+1)}$, and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.
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