Preface

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

ix