@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}},
}