Intersection Exponents for Planar Brownian Motion
نویسندگان
چکیده
منابع مشابه
Multiple intersection exponents for planar Brownian motion
Let p ≥ 2, n1 ≤ · · · ≤ np be positive integers and B 1 , . . . , B n1 ; . . . ;B p 1 , . . . , B np be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a p-fold intersection exponent ςp(n1, . . . , np), as the exponential rate of decay of the probability that the packets ⋃ni j=1 B i j [0, t ], i = 1, . . . , p, have no joint intersection. The ...
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We derive properties concerning all intersection exponents for planar Brownian motion and we deene generalized exponents that loosely speaking correspond to non-integer numbers of Brownian paths. Some of these properties lead to general conjectures concerning the exact value of these exponents.
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We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let B and B′ be independent planar Brownian motions started from distinct points, and define the exponent ξ(1, λ) by E [ P [ B[0, t] ∩B[0, t] = ∅ ∣∣ B[0, t] ]λ ] ≈ t, t → ∞. Then the mapping λ 7→ ξ(1, λ) is real analytic in (0,∞). The same result is proved for the exponents ξ(k, λ) where k is a pos...
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In this review paper, we rst discuss some open problems related to two-dimensional self-avoiding paths and critical percolation. We then review some closely related results (joint work with Greg Lawler and Oded Schramm) on critical exponents for two-dimensional simple random walks, Brownian motions and other conformally invariant random objects.
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ژورنال
عنوان ژورنال: The Annals of Probability
سال: 1999
ISSN: 0091-1798
DOI: 10.1214/aop/1022677543