Phase transition in reinforced random walk and RWRE on trees. (English) Zbl 0648.60077

Pemantle, Robin

Ann. Probab. 16, No. 3, 1229-1241 (1988).

A random walk on an infinite tree is given a particular type of positive feedback, so that edges already traversed are more likely to be traversed in the future. Using exchangeability theory, the process is shown to be equivalent to a random walk in a random environment, that is to say, a mixture of Markov chains; this is achieved via Pólya’s urn scheme.
Criteria are given to determine whether a random walk in a random environment is transient or recurrent. These criteria apply to show that the reinforced random walk can vary from transient to recurrent, depending on the value of an adjustable parameter measuring the strength of the feedback. The value of the parameter at the phase transition is calculated. A crucial role in distinguishing between recurrence and transience is played by Chernoff’s theory of large deviations.

Reviewer: J.L.Teugels

MSC:

60G50	Sums of independent random variables; random walks
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F10	Large deviations

Keywords:

random walk on an infinite tree; exchangeability theory; Pólya’s urn scheme; random walk in a random environment; recurrence; transience; Chernoff’s theory of large deviations