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Range-Controlled Random Walks
Physical Review Letters ( IF 8.1 ) Pub Date : 2023-05-30 , DOI: 10.1103/physrevlett.130.227101
L Régnier 1 , O Bénichou 1 , P L Krapivsky 2, 3
Affiliation  

We introduce range-controlled random walks with hopping rates depending on the range N, that is, the total number of previously distinct visited sites. We analyze a one-parameter class of models with a hopping rate Na and determine the large time behavior of the average range, as well as its complete distribution in two limit cases. We find that the behavior drastically changes depending on whether the exponent a is smaller, equal, or larger than the critical value, ad, depending only on the spatial dimension d. When a>ad, the forager covers the infinite lattice in a finite time. The critical exponent is a1=2 and ad=1 when d2. We also consider the case of two foragers who compete for food, with hopping rates depending on the number of sites each visited before the other. Surprising behaviors occur in 1D where a single walker dominates and finds most of the sites when a>1, while for a<1, the walkers evenly explore the line. We compute the gain of efficiency in visiting sites by adding one walker.

中文翻译:


范围控制随机游走



我们介绍 d 乌瑟尔 a nge 控制的随机游走,跳跃率取决于范围 N ,即之前访问过的不同站点的总数。我们分析具有跳跃率的单参数类模型 Na 并确定平均范围的大时间行为,及其在两种极限情况下的完整分布。我们发现,根据指数 a 是否小于、等于或大于临界值,行为会发生巨大变化, ad ,仅取决于空间维度 d。什么时候 a>ad ,觅食者在有限时间内覆盖无限格子。临界指数是 a1=2ad=1 什么时候 d2 。我们还考虑了两个争夺食物的觅食者的情况,跳跃率取决于彼此之前访问过的站点数量。令人惊讶的行为发生在一维中,其中单个步行者占主导地位,并在以下情况下找到了大多数站点 a>1 ,同时对于 a<1 ,步行者均匀地探索线路。我们通过添加一名步行者来计算访问站点的效率增益。
更新日期:2023-05-31
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