## On the continuity of Lyapunov exponents of random walk in random potential.(English)Zbl 1365.60048

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 M. P. W. Zerner [Ann. Appl. Probab. 8, No. 1, 246–280 (1998; Zbl 0938.60098)] and 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 $$\mathbb{R}^d$$.

### MSC:

 60G50 Sums of independent random variables; random walks 60F05 Central limit and other weak theorems

### Keywords:

random walk; random potential; continuity; Lyapunov exponents

### Citations:

Zbl 0938.60098; Zbl 1193.60033
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