Freezing Freezing, defreezing and reduction to two key lemmas

5.1 Freezing, defreezing and reduction to two key lemmas

5.1.1 Freezing

We begin by deriving a preliminary upper bound for the expectation in Proposition 3.4 given by E ν ρ ,0 exp Z t ds V Z s 5.1 with V η, x = α κ S T φ η, x = α κ X y ∈Z 3 p 6T 1[ κ] x, yη y − ρ, 5.2 where, as before, T is a large constant. To this end, we divide the time interval [0, t] into ⌊tR κ ⌋ intervals of length R κ = Rκ 2 5.3 with R a large constant, and “freeze” the exclusion dynamics ξ t κ t ≥0 on each of these intervals. As will become clear later on, this procedure allows us to express the dependence of 5.1 on the random walk X in terms of objects that are close to integrals over occupation time measures of X on time intervals of length R κ . We will see that the resulting expression can be estimated from above by “defreezing” the exclusion dynamics. We will subsequently see that, after we have taken the limits t → ∞, κ → ∞ and T → ∞, the resulting estimate can be handled by applying a large deviation principle for the space-time rescaled occupation time measures in the limit as R → ∞. The latter will lead us to the polaron term. Ignoring the negligible final time interval [ ⌊tR κ ⌋R κ , t], using Hölder’s inequality with p, q 1 and 1 p + 1q = 1, and inserting 5.2, we see that 5.1 may be estimated from above as E ν ρ ,0 exp Z ⌊tR κ ⌋R κ ds V Z s = E ν ρ ,0 ‚ exp – α κ ⌊tR κ ⌋ X k=1 Z kR κ k−1R κ ds X y ∈Z 3 p 6T 1[ κ] X s , y ξ s κ y − ρ ™Œ ≤ E 1 R, αq t 1 q E 2 R, αp t 1 p 5.4 with E 1 R, α t = E 1 R, α κ, T ; t = E ν ρ ,0 ‚ exp – α κ ⌊tR κ ⌋ X k=1 Z kR κ k−1R κ ds X y ∈Z 3 p 6T 1[ κ] X s , y ξ s κ y − p 6T 1[ κ]+ s −k−1Rκ κ X s , y ξ k−1Rκ κ y ™Œ 5.5 and E 2 R, α t = E 2 R, α κ, T ; t = E ν ρ ,0 ‚ exp – α κ ⌊tR κ ⌋ X k=1 Z kR κ k−1R κ ds X y ∈Z 3 p 6T 1[ κ]+ s −k−1Rκ κ X s , y ξ k−1Rκ κ y − ρ ™Œ . 5.6 2108 Therefore, by choosing p close to 1, the proof of the upper bound in Proposition 3.4 reduces to the proof of the following two lemmas. Lemma 5.1. For all R, α 0, lim sup t, κ,T →∞ κ 2 t log E 1 R, α κ, T ; t ≤ 0. 5.7 Lemma 5.2. For all α 0, lim sup R →∞ lim sup t, κ,T →∞ κ 2 t log E 2 R, α κ, T ; t ≤ 6 α 2 ρ1 − ρ 2 P 3 . 5.8 Lemma 5.1 will be proved in Section 5.1.2, Lemma 5.2 in Sections 5.1.3–5.3.3.

5.1.2 Proof of Lemma 5.1