2,881 research outputs found
Central limit theorem for first-passage percolation time across thin cylinders
We prove that first-passage percolation times across thin cylinders of the
form obey Gaussian central limit theorems as
long as grows slower than . It is an open question as to
what is the fastest that can grow so that a Gaussian CLT still holds.
Under the natural but unproven assumption about existence of fluctuation and
transversal exponents, and strict convexity of the limiting shape in the
direction of , we prove that in dimensions 2 and 3 the CLT holds
all the way up to the height of the unrestricted geodesic. We also provide some
numerical evidence in support of the conjecture in dimension 2.Comment: Final version, accepted in Probability Theory and Related Fields. 40
pages, 7 figure
50 years of first passage percolation
We celebrate the 50th anniversary of one the most classical models in
probability theory. In this survey, we describe the main results of first
passage percolation, paying special attention to the recent burst of advances
of the past 5 years. The purpose of these notes is twofold. In the first
chapters, we give self-contained proofs of seminal results obtained in the '80s
and '90s on limit shapes and geodesics, while covering the state of the art of
these questions. Second, aside from these classical results, we discuss recent
perspectives and directions including (1) the connection between Busemann
functions and geodesics, (2) the proof of sublinear variance under 2+log
moments of passage times and (3) the role of growth and competition models. We
also provide a collection of (old and new) open questions, hoping to solve them
before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with
expanded and additional material. Small typos corrected throughou
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