What happens if a drunkard is asked to walk a full circle around a lake in a short time?

Abstract

A random walker during time t moves distance that scales as t 1/2 . This critical exponent of ν = 1/2 shows up in innumerable variety of circumstances in physics, sometimes in disguise, such as, e.g., the decay of probability distribution for the classical gambler's ruin problem (first passage time distribution) as ∼ t −3/2 is in fact ∼ t −(2+d)ν .Kardar-Parisi-Zhang model of a growing surface gives the most widely known example of fundamentally different and independent scaling ∼ t 2/3 (see, e.g. textbook [1, section 9.6]).