Energy dynamics in the Sinai model

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We study the time-dependent potential energy W( t)= U( x(0))− U( x( t)) of a particle diffusing in a one-dimensional random force field (the Sinai model). Using the real space renormalization group method (RSRG), we obtain the exact large time limit of the probability distribution of the scaling variable w=W(t)/(T ln t) . This distribution exhibits a non-analytic behaviour at w=1. These results are extended to a small non-zero applied field. Using the constrained path integral method, we moreover compute the joint distribution of energy W( t) and position x( t) at time t. In presence of a reflecting boundary at the starting point, with possibly some drift in the + direction, the RSRG very simply yields the one time and aging two-time behaviour of this joint probability. It exhibits differences in behaviour compared to the unbounded motion, such as analyticity. Relations with some magnetization distributions in the 1D spin glass are discussed.