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PRODID:UW-Madison-Physics-Events
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SEQUENCE:0
UID:UW-Physics-Event-7791
DTSTART:20220808T180000Z
DURATION:PT1H0M0S
DTSTAMP:20240914T211520Z
LAST-MODIFIED:20220708T190603Z
LOCATION:4272 Chamberlin Hall
SUMMARY:Random Entanglement and History-Dependent Random Sequences\, T
hesis Defense\, Gage Bonner\, Physics PhD Graduate Student
DESCRIPTION:This thesis is comprised of two main parts\, each concerne
d with a different stochastic process. In the first section\, we consi
der a two-dimensional reflecting Brownian motion in a bounded region d
ivided into two halves by a wall with three or more small windows. In
the limit of small windows\, we appeal to the narrow escape problem to
construct a Markov chain on the fundamental groupoid of the region. T
aking the transition probabilities between windows as inputs\, our Mar
kov chain can be cast as a random walk on a regular language. We obtai
n a law of large numbers as well as a central limit theorem for this p
rocess in which the constants appearing in the limit theorems are expr
essed in terms of a coupled system of quadratic equations. Our result
requires the solution of a simpler problem than those seen previously
in the literature\, and requires less assumptions.

\n

\nIn the
second section\, we consider several history-dependent sequences whic
h have attracted recent interest. We primarily study the Ulam-Kac adde
r\, a sequence for which very little is known explicitly except for it
s first two moments\, which have been computed in some generality. Our
main contribution is to compute the asymptotic behavior of all moment
s and obtain bounds on their size. We also provide a semi-analytic for
mulation of the moment problem which allows them to be computed direct
ly. We discuss many combinatorial interpretations of this sequence and
others which\, in particular\, lead to a novel formula for the first
passage times of a related sequence. Connections of these sequences to
other areas of mathematics and physics are explored.
URL:https://www.physics.wisc.edu/events/?id=7791
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