This book aims to provide a systematic account for some recent progress on
the large deviations arising from the area of sample path intersections, including
calculation of the tail probabilities of the intersection local times, the ranges and
the intersections of the ranges of random walks and Brownian motions. The phrase
“related topics” appearing in the title of the book mainly refers to the weak law
and the law of the iterated logarithm for these models. The former is the reason
for certain forms of large deviations known as moderate deviations; while the latter
appears as an application of the moderate deviations.
Quantities measuring the amount of self-intersection of a random walk, or of
mutual intersection of several independent random walks have been studied inten-
sively for more than twenty years; see e.g. [57], [59], [124], [125], [116], [22],
[131], [86], [135][136], [17], [90], [11], [10], [114]. This research is often moti-
vated by the role that these quantities play in renormalization group methods for
quantum field theory (see e.g. [78], [51], [52], [64]); our understanding of polymer
models (see e.g. [134], [19],[96], [98] [162], [165], [166],[167],[63], [106], [21],
[94], [93]); or the analysis of stochastic processes in random environments (see e.g.
[107], [111],[43], [44] [82], [95], [4], [42] [79], [83]).
Sample path intersection is also an important subject within the probability
field. It has been known ([48], [138], [50]) that sample path intersections have a
deep link to the problems of cover times and thick points through tree-encoding
techniques. In addition, it is impossible to write a book on sample path intersec-
tion without mentioning the influential work led by Lawler, Schramm and Werner
([118], [119], [120], [117]) on the famous intersection exponent problem and on
other Brownian sample path properties in connection to the Stochastic Loewner
Evolution, which counts as one of the most exciting developments made in the
fields of probability in recent years.
Contrary to the behavior patterns investigated by Lawler, Schramm and Werner,
where the sample paths avoid each other and are loop-free, most of this book is
concerned with the probability that the random walks and Brownian motions in-
tersect each other or themselves with extreme intensity. When these probabilities
decay at exponential rates, the problem falls into the category of large deviations.
In recent years, there has been some substantial input about the new tools and
new ideas for this subject. The list includes the method of high moment asymp-
totics, sub-additivity created by moment inequality, and the probability in Banach
space combined with the Feynman-Kac formula. Correspondent to the progress in
methodology, established theorems have been accumulated into a rather complete
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