论文信息 - SCALING LIMIT OF THE PRUDENT WALK

SCALING LIMIT OF THE PRUDENT WALK

We describe the scaling limit of the nearest neighbour prudent walk on $Z^2$, which performs steps uniformly in directions in which it does not see sites already visited. We show that the scaling limit is given by the process $Z_u = \int_0^{3u/7} ( \sigma_1 1_{W(s)\geq 0}\vec{e}_1 + \sigma_2 1_{W(s)\geq 0}\vec{e}_2 ) ds$, $u \in [0,1]$, where $W$ is the one-dimensional Brownian motion and $\sigma_1, \sigma_2$ two random signs. In particular, the asymptotic speed of the walk is well-defined in the $L^1$-norm and equals 3/7.

[1] J. Debierre,et al. Self-directed walk: a Monte Carlo study in three dimensions , 1987 .

[2] Omer Angel,et al. Random Walks that Avoid Their Past Convex Hull , 2002 .

[3] Anthony J. Guttmann,et al. Prudent Self-Avoiding Walks , 2008, Entropy.

[4] A. Guttmann. Some solvable, and as yet unsolvable, polygon and walk models , 2006 .

[5] S. Resnick. Heavy-Tail Phenomena: Probabilistic and Statistical Modeling , 2006 .

[6] Péter Major,et al. The approximation of partial sums of independent RV's , 1976 .

[7] Gregory F. Lawler,et al. Random Walk: A Modern Introduction , 2010 .

[8] S. M. Shcolnick,et al. On the ratio of independent stable random variables , 1985 .

[9] Uwe Schwerdtfeger. Exact solution of two classes of prudent polygons , 2010, Eur. J. Comb..

[10] L. A. Shepp,et al. Symmetric random walk , 1962 .

[11] Enrica Duchi. On some classes of prudent walks , 2005 .

[12] D J Klein,et al. Directed self-avoiding walks in random media. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13] Anthony J. Guttmann,et al. Prudent walks and polygons , 2008 .

[14] P. Major,et al. An approximation of partial sums of independent RV'-s, and the sample DF. I , 1975 .

[15] Mireille Bousquet-Mélou,et al. Families of prudent self-avoiding walks , 2008, J. Comb. Theory, Ser. A.

[16] William Feller,et al. An Introduction to Probability Theory and Its Applications , 1951 .