Large Deviations for a random walk in random environment - Greven, Andreas; den Hollander, Frank (Annals of Probability, 1994)
Let ω=(px)x∈Z" role="presentation">ω=(px)x∈ℤ be an i.i.d. collection of (0, 1)-valued random variables. Given ω" role="presentation">ω, let (Xn)n≥0" role="presentation">(Xn)n≥0 be the Markov chain on Z" role="presentation">ℤ defined by X0=0" role="presentation">X0=0 and Xn+1=Xn+1(resp.Xn−1)" role="presentation">Xn+1=Xn+1(resp.Xn−1) with probability pXn(resp⁡.1−pXn)" role="presentation">pXn(resp.1−pXn). It is shown that Xn/n" role="presentation">Xn/n satisfies a large deviation principle with a continuous rate function, that is, limn→∞1nlog⁡Pω(Xn=⌊θnn⌋)=−I(θ)ω−a.s.forθn→∈[−1,1]." role="presentation">limn→∞1nlogPω(Xn=⌊θnn⌋)=−I(θ)ω−a.s.forθn→∈[−1,1]. First, we derive a representation of the rate function I" role="presentation">I in terms of a variational problem. Second, we solve the latter explicitly in terms of random continued fractions. This leads to a classification and qualitative description of the shape of I" role="presentation">I. In the recurrent case I" role="presentation">I is nonanalytic at θ=0" role="presentation">θ=0. In the transient case I" role="presentation">I is nonanalytic at θ=−θc,0,θc" role="presentation">θ=−θc,0,θc for some θc≥0" role="presentation">θc≥0, with linear pieces in betwee