The intersection exponent for simple random walk (Q2709848)

Let \(S=(S_n)_{n\in\mathbb N_0}\) be a simple random walk in \(\mathbb Z^d\) (with \(d=2\) or \(d=3\)), and let \(S^1,S^2,\dots\) be independent copies of \(S\). Given \(\lambda\in[0,\infty)\) and \(k\in\mathbb N\), the (generalized) intersection exponent \(\xi_d(\lambda,k)\) is defined by the asymptotic relation \(E(Y_n^\lambda)\approx n^{-\xi_d(\lambda,k)}\) as \(n\to\infty\), where \(\approx\) means logarithmic equivalence, and \(Y_n\) is defined as \(P(S[0,n^2]\cap(S^1[0,n^2]\cup\dots\cup S^k[0,n^2]) |S[0,n^2])\). For integers \(\lambda\), \(E(Y_n^\lambda)\) is just the probability that the union of \(n^2\)-step paths of \(\lambda\) walks do not intersect the union of \(k\) independent other ones, which explains the notion of an intersection exponent (actually, it is a non-intersection exponent). For \(\lambda=0\), we interprete \(Y_n^0\) as the indicator on \(\{Y_n>0\}\). The exponent \(\xi_d(0,k)\) plays an important role in the fine analysis of the boundary of the path \(S[0,n^2]\). Dimensions two and three are the only dimensions in which \(\xi_d\) is non-trivial. NEWLINENEWLINENEWLINEThe two main results of the paper are: (1) It is shown that the analogously defined intersection exponents for Brownian motions coincide with the above exponents for simple random walk, and (2) the relation \(E(Y_n^\lambda)\approx n^{-\xi_d(\lambda,k)}\) is strengthened to \(E(Y_n^\lambda)\asymp n^{-\xi_d(\lambda,k)}\), where \(\asymp\) means that the quotient of the two sides is bounded and bounded away from zero. One main tool for deriving the latter result is a joint Skorokhod embedding of several random walks on a Brownian probability space. The results of the present paper are used elsewhere by one of the authors to analyse the multifractal spectrum of harmonic measure on a random walk path. NEWLINENEWLINENEWLINEThe intersection exponents belong to a class of models from statistical physics that are easy to define, but exhibit deeply hidden and not yet understood critical phenomena. In dimension two, the precise values of the exponents have been conjectured by physicists on base of (conjectured) conformal invariance properties of the scaling limit. There is hope that in near future mathematicians will be able to define candidates for scaling limit processes for various of these models, including the intersection exponents. The importance of the present paper lies in showing that, for intersection exponents, the continuum limit gives the correct values.