Random Walk in Periodic Environment

We consider a special case of random walk in random environment (RWRE) on Z^d where the environment is periodic (RWPE). Under natural conditions, we show that law of large numbers and central limit theorem holds. In the ballistic nearest neighbour reversible case, we prove that the angle between the asymptotic direction of the RWPE and the average negative gradient of the potential function of the reversible environment is less than pi/2, that is, the potential cannot increase asymptotically along the trajectory of the RWPE. But this angle can be close to pi/2.

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