Journal
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
Volume 2020, Issue 5, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/1742-5468/ab7c5f
Keywords
extreme value statistics; stochastic processes
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Funding
- German Science Foundation (DFG) [HA 3169/8-1]
- DFG through its Major Research Instrumentation Programme [INST 184/108-1 FUGG, INST 184/157-1 FUGG]
- Ministry of Science and Culture (MWK) of the Lower Saxony State
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We study the statistical properties of the convex hull of a planar run-and-tumble particle (RTP), also known as the 'persistent random walk', where the particle/walker runs ballistically between tumble events at which it changes its direction randomly. We consider two different statistical ensembles where we either fix (i) the total number of tumblings n or (ii) the total duration t of the time interval. In both cases, we derive exact expressions for the average perimeter of the convex hull and then compare to numerical estimates finding excellent agreement. Further, we numerically compute the full distribution of the perimeter using Markov chain Monte Carlo techniques, in both ensembles, probing the far tails of the distribution, up to a precision smaller than 10(-100). This also allows us to characterize the rare events that contribute to the tails of these distributions.
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