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I'm curious if there is a known asymptotic scaling for the return-to-origin (i.e. recurrence) probability for a random on $Z^d$ as a function of $d$?

Mathworld gives the recurrence probability: http://mathworld.wolfram.com/PolyasRandomWalkConstants.html

$p(d) = 1-\frac{1}{u(d)} = 1- \left(\int_{t=0}^{\infty}I_0(\frac{t}{d})^de^{-t} dt\right)^{-1}$

Where $I_0$ is a modified Bessel function of the first kind: http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html

Taking a wild guess, I found that a quartic fit seems to work well for values of $p(d)$ for $d = 1$ to $8$: http://www.wolframalpha.com/input/?i=quartic+fit+%7B3%2C+0.340537%7D%2C%7B4%2C+0.193206%7D%2C%7B5%2C+0.135178%7D%2C%7B6%2C+0.104715%7D%2C%7B7%2C+0.0858449%7D%2C%7B8%2C+0.0729126%7D

Also, what's the best way to approximate the integral expression necessary to calculate $p(d)$ if we only need $k$ digits of precision?


Sorry folks: Pólya's Random Walk Constants at infinity

I should have done a more thorough job checking for previous questions about this topic. =(

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  • $\begingroup$ corrected parentheses in formula $\endgroup$ – Gerald Edgar Dec 26 '12 at 15:32
  • $\begingroup$ Montroll's paper referred in your link to wolframalpha has asymptotics when d tends to infinity. $\endgroup$ – Alexandre Eremenko Dec 26 '12 at 16:32
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Asymptotic in the sense $d \to \infty$? From Maple I get: $$ p(d) = \frac{1}{2 d} + \frac{1}{2 d^{2}} + \frac{7}{8 d^{3}} + \frac{35}{16 d^{4}} + O \Bigl(d^{-5}\Bigr) $$ as $d \to \infty$.

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