On the continuity of Lyapunov exponents of random walk in random potential
The paper considers a simple random walk on the \(d\)-dimensional \((d\geq 3)\) cubic lattice. Independently of the random walk, a family of i.i.d. random variables taking values in \([0,\infty )\) is given, which are called potentials. Let \(F\) be the distribution function of the potential. Using \(F\), following the results of \textit{M. P. W. Zerner} [Ann. Appl. Probab. 8, No. 1, 246--280 (1998; Zbl 0938.60098)] and \textit{M. Flury} [Stochastic Processes Appl. 117, No. 5, 596--612 (2007; Zbl 1193.60033)], the author introduces \(\alpha _F(x)\) and \(\beta _F(x)\), the corresponding quenched and annealed Lyapunov exponents. He studies the continuity of these exponents with respect to the law of the potential, assuming independence. Let \(\mathcal{D}\) be the set of distribution functions, \(\mathcal{D}_1\) the subset of \(\mathcal{D}\) containing all distribution functions with finite mean and \((F_n)\) a sequence of distribution functions. It is shown that if \(F_n\) tends to \(F\), then \(\lim _{n\to \infty }\alpha _{F_n}(x)=\alpha _F(x)\) in \(\mathcal{D}_1\) and \(\lim _{n\to \infty }\beta _{F_n}(x)=\beta _F(x)\) in \(\mathcal{D}\) for all \(x\in \mathbb{R}^d\), and the convergence is uniform on any compact subset of \(\mathbb{R}^d\).
scientific article; zbMATH DE number 6673487