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A random walk of length k on a possibly infinite graph G with a root 0 is a stochastic process with random variables ,, …, such that = and + is a vertex chosen uniformly at random from the neighbors of .
In this case it turns out that there the intersections are only local. A calculation shows that if one takes a random walk of length n, its loop-erasure has length of the same order of magnitude, i.e. n. Scaling accordingly, it turns out that loop-erased random walk converges (in an appropriate sense) to Brownian motion as n goes to infinity ...
The actual random walk obeys a stochastic equation of motion, but its probability density function (PDF) obeys a deterministic equation. PDFs of random walks can be formulated in terms of the (discrete in space) master equation [1] [12] [13] and the generalized master equation [3] or the (continuous in space and time) Fokker Planck equation [37] and its generalizations. [10]
Maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy.
The probability that a random walk on a Bethe lattice of degree starting at a given vertex eventually returns to that vertex is given by . To show this, let P ( k ) {\displaystyle P(k)} be the probability of returning to our starting point if we are a distance k {\displaystyle k} away.
In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times. [1] [2] [3] More generally it can be seen to be a special case of a Markov renewal process.
A Lévy flight is a random walk in which the step-lengths have a stable distribution, [1] a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases ...
Branching random walk; Brownian web; C. Chan–Karolyi–Longstaff–Sanders process; Continuous-time random walk; D. Double Fourier sphere method; G. Gambler's ruin; H.