We study a general class of translation invariant quantum Markov evolutions for a particle on $\bbZ^d$. The evolution consists of free flow, interrupted by scattering events. We assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution. It is shown that the particle position diffuses in the long time limit. This generalizes standard results about central limit theorems for classical (non-quantum) Markov processes.Comment: 21 pages. Minor corrections of previous versio

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number b ∈ N and a segment number s ∈ N. When b ≤ s previous work [27] has established that the model exhibits strong disorder for all positive values of the inverse temperature β, and thus weak disorder reigns only for β = 0 (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature β ≡ β n vanishes at an appropriate rate as the size n of the system grows. Our analysis requires separate treatment for the cases b < s and b = s. In the case b < s we prove that when the inverse temperature is taken to be of the form β n = β(b/s) n/2 for β > 0, the normalized partition function of the system converges weakly as n → ∞ to a distribution L( β) depending continuously on the parameter β. In the case b = s we find a critical point in the behavior of the model when the inverse temperature is scaled as β n = β/n; for an explicitly computable critical value κ b > 0 the variance of the normalized partition function converges to zero with large n when β ≤ κ b and grows without bound when β > κ b . Finally, we prove a central limit theorem for the normalized partition function when β ≤ κ b .

I discuss models for a continuum directed random polymer in a disordered environment in which the polymer lives on a fractal called the diamond hierarchical lattice, a self-similar metric space forming a network of interweaving pathways. This fractal depends on a branching parameter b ∈ N and a segmenting number s ∈ N. For s > b my focus is on random measures on the set of directed paths that can be formulated as a subcritical Gaussian multiplicative chaos. This path measure is analogous to the continuum directed random polymer introduced by Alberts, Khanin, Quastel [Journal of Statistical Physics 154, 305-326 (2014)].

I study a model for a massive one-dimensional particle in a singular periodic potential that is receiving kicks from a gas. The model is described by a Lindblad equation in which the Hamiltonian is a Schrödinger operator with a periodic δ-potential and the noise has a frictionless form arising in a Brownian limit. I prove that an emergent Markov process in a semi-classical limit governs the momentum distribution in the extended-zone scheme. The main result is a central limit theorem for a time integral of the momentum process, which is closely related to the particle's position. When normalized by t 5 4 , the integral process converges to a time-changed Brownian motion whose rate depends on the momentum process. The scaling t 5 4 contrasts with t 3 2 , which would be expected for the case of a smooth periodic potential or for a comparable classical process. The difference is a wave effect driven by Bragg reflections that occur when the particle's momentum is kicked near the half-spaced reciprocal lattice.where the Hilbert spaces H φ are canonically identified with L 2 [−π, π) , and the restriction of H to the φ-fiber is a self-adjoint operator H φ . The operators H φ have a complete set of eigenvectors ψ n,φ , n ∈ N with eigenvalues E n,φ satisfying 0 ≤ E n,φ ≤ E n+1,φ , E n,φ −→ ∞ as n −→ ∞.

In this article, we study a second-order expansion for the effect induced on a large quantum particle which undergoes a single scattering with a low-mass particle via a repulsive point interaction. We give an approximation with third-order error in λ to the map G → Tr 2 [(I ⊗ρ)S * λ (G⊗I )S λ ], where G ∈ B(L 2 (R n )) is a heavy-particle observable, ρ ∈ B 1 (R n ) is the density matrix corresponding to the state of the light particle, λ = m M is the mass ratio of the light particle to the heavy particle, S λ ∈ B(L 2 (R n ) ⊗ L 2 (R n )) is the scattering matrix between the two particles due to a repulsive point interaction, and the trace is over the light-particle Hilbert space. The third-order error is bounded in operator norm for dimensions one and three using a weighted operator norm on G.