On the critical exponent for random walk intersections. (English) Zbl 0714.60057
Summary: The exponent \(\zeta_ d\) for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that \(1/2<\zeta_ 2\leq 3/4\). Monte Carlo simulations are done to suggest \(\zeta_ 2=0.61..\). and \(\zeta_ 3\approx 0.29\).
References:
| [1] | K. Burdzy and G. Lawler, Non-intersection exponents for Brownian paths, I and II, preprints. · Zbl 0685.60080 |
| [2] | B. Duplantier, Intersections of random walks. A direct renormalization approach,Commun. Math. Phys. 117:279-330 (1988). · Zbl 0652.60073 · doi:10.1007/BF01223594 |
| [3] | B. Duplantier and K.-H. Kwon, Conformai invariance and intersections of random walks,Phys. Rev. Lett. 61:2514-2517 (1988). · doi:10.1103/PhysRevLett.61.2514 |
| [4] | P. Erdös and S. J. Taylor, Some intersection properties of random walk paths,Acta Math. Sci. Hung. 11:231-248 (1960). · Zbl 0096.33302 · doi:10.1007/BF02020942 |
| [5] | W. Feller,An Introduction to Probability Theory and its Applications, Vol. I, 3rd ed. (Wiley, New York, 1968). · Zbl 0155.23101 |
| [6] | H. Kesten, Hitting probabilities of random walks on ? d ,Stochastic Processes Appl. 25:165-184 (1987). · Zbl 0626.60067 · doi:10.1016/0304-4149(87)90196-7 |
| [7] | G. Lawler, The probability of intersection of random walks in four dimensions,Commun. Math. Phys. 86:539-554 (1982). · Zbl 0502.60057 · doi:10.1007/BF01214889 |
| [8] | G. Lawler, Intersections of random walks in four dimensions II,Commun. Math. Phys. 97:583-594 (1985). · Zbl 0585.60069 · doi:10.1007/BF01221219 |
| [9] | G. Lawler, Intersections of random walks with random sets,Israel J. Math. (1989). · Zbl 0679.60071 |
| [10] | G. Lawler, Estimates for differences and Harnack’s inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous increments, to appear. · Zbl 0774.39004 |
| [11] | F. Spitzer,Principles of Random Walk, 2nd ed. (Springer-Verlag, New York, 1976). · Zbl 0359.60003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
