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Random Walks and Forest Fire Models: Recurrence, Transience and Phase Transitions

Bhattacharya, Satyaki LU (2026)
Abstract
Random Walk is one of the holy grails of Probability Theory. This literature is born from a simple, nagging
question: what happens to the drunkard’s walk if his steps are no longer equal? The classical theory of random
walks, a cornerstone of probability, offers elegant answers for the simple, symmetric case. But life and many
stochastic models are rarely so uniform.
We step into this gap by studying the Rademacher walk, defined by the sum Sn =
∑n a_iX_i
i=1 , where the
step sizes (ai) are a fixed, deterministic sequence and the Xi are independent Rademacher random variables. Our
central concern is the ancient dichotomy of recurrence and transience: does the path return to a neighbourhood
of the origin... (More)
Random Walk is one of the holy grails of Probability Theory. This literature is born from a simple, nagging
question: what happens to the drunkard’s walk if his steps are no longer equal? The classical theory of random
walks, a cornerstone of probability, offers elegant answers for the simple, symmetric case. But life and many
stochastic models are rarely so uniform.
We step into this gap by studying the Rademacher walk, defined by the sum Sn =
∑n a_iX_i
i=1 , where the
step sizes (ai) are a fixed, deterministic sequence and the Xi are independent Rademacher random variables. Our
central concern is the ancient dichotomy of recurrence and transience: does the path return to a neighbourhood
of the origin infinitely often, or does it wander off forever?
We discover that the answer is a delicate balance between the growth of the step sequence and the geometry of
the path. In one dimension, we find a lower bound of the threshold: if the steps grow like n

α+o(1) for α > 1/2,
the walk is transient, and this bound is exact. But we also construct sequences that grow arbitrarily fast yet still
produce a weakly recurrent walk—a finding that challenges intuition. Moreover, we produce arbitrarily slowly
growing sequences which result in transience. We also extend the work on 2− dimensions and prove similar
results. A parallel study of a forest fire model with delays shows how relaxing standard assumptions can produce
entirely new phenomena, like an “infinite fire.” A parameter of fire spreading time plays a serious role in phase
transition. In addition we find how quickly fires spread. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Prof. Janson, Svante, Uppsala University, Sweden.
organization
publishing date
type
Thesis
publication status
published
subject
pages
174 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Lecture Hall MH:Hörmander, Centre of Mathematical Sciences, Märkesbacken 4, Faculty of Engineering LTH, Lund University, Lund.
defense date
2026-09-09 13:00:00
ISBN
978-91-90202-203
978-91-90202-210
language
English
LU publication?
yes
id
395b381a-3e6f-43b9-9ef6-76949d54c84c
date added to LUP
2026-05-05 15:02:31
date last changed
2026-08-19 08:52:33
@phdthesis{395b381a-3e6f-43b9-9ef6-76949d54c84c,
  abstract     = {{Random Walk is one of the holy grails of Probability Theory. This literature is born from a simple, nagging<br/>question: what happens to the drunkard’s walk if his steps are no longer equal? The classical theory of random<br/>walks, a cornerstone of probability, offers elegant answers for the simple, symmetric case. But life and many<br/>stochastic models are rarely so uniform.<br/>We step into this gap by studying the Rademacher walk, defined by the sum Sn =<br/>∑n  a_iX_i<br/>i=1 , where the<br/>step sizes (ai) are a fixed, deterministic sequence and the Xi are independent Rademacher random variables. Our<br/>central concern is the ancient dichotomy of recurrence and transience: does the path return to a neighbourhood<br/>of the origin infinitely often, or does it wander off forever?<br/>We discover that the answer is a delicate balance between the growth of the step sequence and the geometry of<br/>the path. In one dimension, we find a lower bound of the threshold: if the steps grow like n<br/><br/>α+o(1) for α &gt; 1/2,<br/>the walk is transient, and this bound is exact. But we also construct sequences that grow arbitrarily fast yet still<br/>produce a weakly recurrent walk—a finding that challenges intuition. Moreover, we produce arbitrarily slowly<br/>growing sequences which result in transience. We also extend the work on 2− dimensions and prove similar<br/>results. A parallel study of a forest fire model with delays shows how relaxing standard assumptions can produce<br/>entirely new phenomena, like an “infinite fire.” A parameter of fire spreading time plays a serious role in phase<br/>transition. In addition we find how quickly fires spread.}},
  author       = {{Bhattacharya, Satyaki}},
  isbn         = {{978-91-90202-203}},
  language     = {{eng}},
  publisher    = {{Centre for Mathematical Sciences, Lund University}},
  school       = {{Lund University}},
  title        = {{Random Walks and Forest Fire Models: Recurrence, Transience and Phase Transitions}},
  url          = {{https://lup.lub.lu.se/search/files/249280029/Thesis_unsigned.pdf}},
  year         = {{2026}},
}