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Slowdown and neutral pockets for a random walk in random environment. (English) Zbl 0947.60095
A random walk in random environment on $$\mathbb Z^d$$ is studied in the case where the drifts are either neutral or ‘point to the right’. The model is defined as follows. Let $$\omega=(\omega_x)_{x\in\mathbb Z^d}$$ denote an i.i.d. sequence of random variables $$\omega_x=(\omega_x(e)) _{e\sim 0}$$ (with marginal distribution $$\mu$$), taking values in the set of probability measures on the set of all the $$2d$$ neighbors of $$x\in\mathbb Z^d$$. Given $$\omega$$, we define a random walk $$(X_n)_{n\in\mathbb N_0}$$ on $$\mathbb Z^d$$, which steps from $$x\in\mathbb Z^d$$ to a neighbor $$x+e$$ of $$x$$ with probability $$\omega_x(e)$$. The walk starts from $$X_0=y$$ under $$P_{\omega,y}$$. Hence, $$\omega$$ serves as a random environment for the random walk $$(X_n)_{n\in\mathbb N_0}$$, and the $$\omega_x$$’s are the local transition probabilities. The consideration of the walk under $$P_{\omega,y}$$ is the so-called ‘quenched’ setting, and under $$P_y=\mu^{\otimes \mathbb Z^d}(d\omega) P_{\omega,y}$$ one has the so-called ‘annealed’ setting.
This model has been studied a lot since the mid-seventies in one dimension, but is much more difficult to handle in dimensions $$d\geq 2$$. In fact, there are many interesting questions yet open (general recurrence and transience criteria are lacking, for instance), and rigorous results are available only since a few years (for example a large-deviation principle by M. Zerner). One of the main reasons for the difficulty is the lack of reversibility of the walk, which amounts to a lack of self-adjointness for related spectral properties.
The author studies the case where the environment has either neutral sites or drifts pointing to the right, more precisely: For some $$\delta>0$$, the probabilities $$\mu(\omega_0(e)=1/2d$$, $$\forall e\sim 0)$$ and $$\mu(\sum_{e\sim 0}\omega_0(e)e\cdot {\text e}_1\geq \delta)$$ are both positive and sum up to one, where $${\text e}_1$$ denotes the first canonical basis vector. The author shows that the random walk drifts to the right under $$P_0$$, and he derives critical large-deviation estimates both in the quenched and the annealed setting for the stopping times $$T_\ell=\inf\{n\in\mathbb N_0\colon X_n\cdot{\text e}_1\geq \ell\}$$. More precisely, he shows that $$-\ell^{d/(d+2)}\log P_0[T^\ell >c\ell]$$ is bounded and bounded away from $$\infty$$ as $$\ell\to\infty$$ for sufficiently large $$c>0$$ (annealed estimates), and he shows that, for $$\mu^{\otimes \mathbb Z^d}$$-almost all $$\omega$$, $$-\ell^{-1}(\log \ell)^{2/d}\log P_{0,\omega}[T^\ell>c\ell]$$ is bounded away from zero and, under some technical assumption, also bounded. The results and heuristics are reminiscent of the problem of Brownian motion in a Poissonian potential, which has been intensively studied by the author in the nineties. The idea for the annealed estimate is that the environment produces around the origin a hole of diameter $$\ell^{1/(d+2)}$$ of neutral sites in which the walk can rest for a large portion of time. In the quenched setting, the ‘resting hole’ has diameter of order $$(\log \ell)^{1/d}$$ and is located within distance $$\ell^{1-\eta}$$ from the origin.
Reviewer: W.König (Berlin)

##### MSC:
 60K40 Other physical applications of random processes 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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