We study the distribution of the number of visits a random walker makes at a given site on a finite lattice with N sites, during a very long walk which visits each ...
Physica A 192 (1993) 465-470 North-Holland
H o w u n i f o r m l y a r a n d o m walker covers a finite lattice Harald Freund and Peter Grassberger Physics Department, University of Wuppertal, W-5600 Wuppertal 1, Germany Received 13 August 1992 Manuscript received in final form 29 September 1992 We study the distribution of the number of visits a random walker makes at a given site on a finite lattice with N sites, during a very long walk which visits each site a large number of times. For regular hypercubic lattices in all dimensions we find normal central limit behavior, but with anomalously large variance in ~2).
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A s expected, this diverges for d---~ 2. F o r d = 2, finally, the same infrared cutoff gives a logarithmic divergence leading to eq. (3). In o r d e r to verify eq. (3) numerically, we show in figs. 1 and 2 results f r o m square lattices of sizes 16 × 16 up to 1024 × 1024. In each case, h was equal to 160, and a large n u m b e r of realizations was a v e r a g e d over in o r d e r to r e d u c e
H. Freund, P. Grassberger I Uniformity of random walker
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fluctuations. In fig. 1 we show the distributions P ( n ) for the smallest and the largest lattice. We see that indeed the width increases with L, but h was sufficiently large so that our asymptotic estimates should apply. The latter was also checked by making some runs with h = 320. The variances (or rather the ratios o'/h) are plotted versus log L in fig. 2. We see the expected linear increase. The fitted slope is 1.29---0.02, to be compared with the prediction 4/'rr = 1.273. While the coefficient 4 / ~ in eq. (3) should be exact, the coefficient 2 / 3 ~ in eq. (12) is only approximate since it depends on the somewhat arbitrary choice of t c. Numerical simulations in 1 dimension gave o / h ~ 0.33L - 1.0. Thus eq. (12) is indeed only qualitatively correct. Finally, we compared the above results with simulations of self-repelling walks. For self-repelling walks the next step is chosen according to a Boltzmann factor ~ exp[-/3n(x)] where/3 is formally an inverse temperature. While 1-dimensional self-repelling walks become trivial at zero temperature, higher-dimensional ones remain non-trivial in this limit in which the problem simplifies somewhat numerically. Results for square lattices with L = 4, 8 . . . . . 2048, and with/3 = o0, are given in fig. 3. Notice that we have measured only at times when the total number of steps was an integer multiple of the lattice size N. Otherwise, we would have self-repelling walks
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an additional contribution to o- due to partially filled layers. We see a beautifully linear behavior, leading to eq. (4). Simulations in d = 1 gave o-~ L ( o - ~ 0.060L for /3 = log 4), while simulations on 3-dimensional cubic lattices gave l i m c ~ o- = 0.234 + 0.002 f o r / 3 = ~. T h e s e results indicate that self-repelling walks lead to a surprisingly uniform c o v e r a g e of 2D lattices (and of higher-dimensional lattices as well). E v e n on a lattice of size 2048 x 2048 the rms. height fluctuation is
