Titlebook: Random Walks on Infinite Groups; Steven P. Lalley Textbook 2023 The Editor(s) (if applicable) and The Author(s), under exclusive license t

凶残

Isoperimetric Inequalities and Amenability,ughly like .., where . ≤ 1 is the so-called . of the walk. When is it the case that this exponential rate . is strictly less than 1? Kesten [.] discovered that the answer depends on a fundamental structural feature of the ambient group: .

笨重

Compact Group Actions and Boundaries,al objects. We have seen that the action of a finitely generated group on its Cayley graph by right multiplication is of fundamental importance in the study of random walks. In this chapter we will investigate group actions on . metric spaces, and show how these can unlock information about the long-time behavior of random walk trajectories.

正常

Hyperbolic Groups,roup. In this chapter, we shall study a large class of groups, the ., first introduced by Gromov (Essays in Group Theory. Springer, New York, 1987), whose geometry forces certain regularities on random walk paths.

微不足道

The Ergodic Theorem,to its initial location .. = 1 is positive. Moreover, for any . the sequence . is a version of the random walk, as its increments .., .., ⋯ are independent and identically distributed with common distribution ., and so the probability that it will ever return to the initial state 1 is also . > 0.

形容词

Ossification

Isoperimetric Inequalities and Amenability,ughly like .., where . ≤ 1 is the so-called . of the walk. When is it the case that this exponential rate . is strictly less than 1? Kesten [.] discovered that the answer depends on a fundamental structural feature of the ambient group: .

Abjure

Myocyte

Martingales,a finite region . is uniquely determined by its values on the boundary .. Consequently, for any finite region . the vector space of real harmonic functions in . is isomorphic to the space of real-valued functions on the boundary .. Is there an analogous characterization of the space of harmonic func

paleolithic

congenial

Hyperbolic Groups,roup. In this chapter, we shall study a large class of groups, the ., first introduced by Gromov (Essays in Group Theory. Springer, New York, 1987), whose geometry forces certain regularities on random walk paths.