Slowdown and neutral pockets for a random walk in random environment.

*(English)*Zbl 0947.60095A 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.

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) |