Archive for August, 2007

Polya Problem

August 19, 2007

Consider a dynamical system under the influence of random noise. In such a general setting, the Polya problem questions the probability of coming back to the place that the system has started. If the configuration space of the system had finite volume then the probability would have been 1 (in the absence of attractors) somewhat in the same setting as ergodicity.

In the random walk frameworks the problem can be formalized easily. Define Q_n(r) as the probability of reaching to position r for the first time after n-steps. The generating function F(r,z) is defined as

F(r,z) = \sum_{n=1}^{\infty} z^n Q_n(r)

The probability of being as position r after n-steps, p_n(r) is following the consistency equation

p_n(r) = \sum_{j=1}^{n} p_{n-j}(0) Q_j(r)

The above equation relates the generating function for p_n(r) to the Q_n generating function

G(r,z) = \delta_{r,0} + F(r,z) G(0,z)

So the total return probability F(0,1)=Q_1(0)+Q_2(0)+ \ldots can be written as

F(0,1)=1-{1 \over {G(0,1)}}

The generating function G can be related to jump vector distribution, f(r), where p_n(r) = \sum f(r-r^{\prime} ) p_{n-1}(r^{\prime}) through convolution

G(r,z) = {1 \over 2 \pi} \int dk {\exp(-i kr) \over 1 - z{\tilde f}(k) }

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