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Considering a "random walk in a random environment" in a topologically closed circuit, we explore the implications of the percolation and sliding transitions for its relaxation modes. A complementary question regarding the "delocalization"…
We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan type behavior…
Using a stochastic quantum approach, we study thermoelectric transport phenomena at low temperatures in disordered electrical systems connected to external baths. We discuss three different models of one-dimensional disordered electrons,…
These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our…
Working within the framework of the covariant perturbation theory, we obtain the coincidence limit of the heat kernel of an elliptic second order differential operator that is applicable to a large class of quantum field theories. The basis…
We use reproducing kernel methods to study various rigidity problems. The methods and setting allow us to also consider the non-positive case.
We consider random walks in dynamic random environments and propose a criterion which, if satisfied, allows to decompose the random walk trajectory into i.i.d. increments, and ultimately to prove limit theorems. The criterion involves the…
We study the propagation of a hole in degenerate (paramagnetic) spin environments. This canonical problem has important connections to a number of physical systems, and is perfectly suited for experimental realization with ultra-cold atoms…
We use discrete-event simulation on a digital computer to study two different models of experimentally realizable quantum walks. The simulation models comply with Einstein locality, are as "realistic" as the one of the simple random walk in…
The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic…
When studying non-symmetric nonlocal operators $$ {\cal L} f(x) = \int_{{\bf R}^d} \left( f(x+z)-f(x)-\nabla f(x)\cdot z 1_{\{|z|\leq 1\}} \right) \frac{\kappa (x, z)}{|z|^{d+\alpha}} d z , $$ where $0<\alpha<2$ and $\kappa (x, z)$ is a…
In discrete energy structure of nanoscale magnets, nonadiabatic transitions at avoided level crossings lead to fundamental processes of dynamics of magnetizations. The thermal environment causes dissipative effects on these processes. In…
We study an unbiased, discrete time random walk on the nonnegative integers, with the origin absorbing. The process has a history-dependent step length: the walker takes steps of length v while in a region which has been visited before, and…
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local…
For the first time, the estimates of the non-centered Weyl-group invariant heat kernel for curved Riemannian spaces and for Opdam-Cherednik Laplacians are studied systematically. We prove sharp estimates for the root system $A_1$ with…
We review recent developments in nonlinear quantum transport through nanostructures and mesoscopic systems driven by thermal gradients or in combination with voltage biases. Low-dimensional conductors are excellent platforms to analyze both…
In this survey article, we review the relation between heat kernels and path integrals. In particular, we review recent results on the approximation of the Wiener measure on compact manifold by measures on (finite-dimensional) spaces of…
In this Colloquium recent advances in the field of quantum heat transport are reviewed. This topic has been investigated theoretically for several decades, but only during the past twenty years have experiments on various mesoscopic systems…
We discuss theoretical issues in radiative rare kaon decays. The interest is twofold: to extract useful short-distance information and understand the underlying dynamics. We emphasize channels where either we can understand non-perturbative…
We calculate explicit speeds for random walks in uniform degenerate random environments. For certain non-uniform random environments, we calculate speeds that are non-monotone.