# PhD Defense Frederik Ravn Klausen

Title: Random Problems in Mathematical Physics

Abstract:

This PhD thesis deals with a number of different problems in mathematical physics with the common thread that they have probabilistic aspects. The problems all stem from mathematical studies of lattice systems in statistical and quantum physics; however beyond that, the selection of the concrete problems is to a certain extent arbitrary. This thesis consists of an introduction and seven papers.

In [Mass], we give a new proof of exponential decay of the truncated two-point correlation functions of the two-dimensional Ising model at the critical temperature in a magnetic field. In [MonCoup], we provide counterexamples to monotonicity properties of the loop O(1) model and the (single, traced, sourceless) random current model. Additionally, we prove that the uniform even subgraph of the (traced, sourceless) double random current model has the law of the loop O(1) model.

In [Kert´esz], we prove strict monotonicity and continuity of the Kert´esz line for the random cluster model in the presence of a magnetic field implemented through a ghost vertex. Furthermore, we give new rigorous bounds that are asymptotically correct in the limit h → 0. In [UEG], we prove that the uniform even subgraph percolates in Zd for d ≥ 2, that the phase transition of the loop O(1) model on Zd is non-trivial and we provide a polynomial lower bound on the correlation functions of both the loop O(1) model and single random current corresponding to a supercritical Ising model on Zd whenever d ≥ 3.

In [MagQW], we introduce a model for quantum walks on Z2 in a random magnetic field where the plaquette fields are i.i.d. random. We prove an a priori estimate and an exponential decay result of the expectations of fractional moments of the Green function. In [Spec], we obtain a representation of generators of Markovian open quantum system with natural locality assumptions as a direct integral of finite range bi-infinite Laurent matrices with finite rank perturbations. We use the representation to calculate the spectrum of some infinite volume open quantum Lindbladians analytically and to prove gaplessness of the spectrum, absence of residual spectrum and a condition for convergence of finite volume spectra to their

infinite volume counterparts. In [OpenLoc], we consider a Markovian open quantum system where the terms in the generator are local. We prove that in the presence of any local dephasing in the system, then any steady state of the system will have exponentially decaying coherences. Furthermore, we prove for a general class of models that includes our motivating examples, that the results holds in expectation for large disorder, that is, a sufficiently strong random potential in the Hamiltonian. That result extends Anderson localization to open quantum systems.

**Supervisor:** Albert H. Werner, University of Copenhagen

**Assessment Committee:**

Professor Jan Philip Solovej, University of Copenhagen

Professor Michael Aizenman, Princeton University

Universitetslektor Jakob Björnberg Chalmers and Göteborg University