Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two parameters q, t in [0,1). Utilizing the Macdonald difference operators we prove several results about observables these processes, including Fredholm determinant formulas for q…

The Kuramoto synchronization model is the reference model for synchronization phenomena in biology (and, to a certain extent, also in other fields). The model is formulated as a dynamical system of interacting plane rotators. Variations of it provide basic models of phenomena beyond synchronization, such as noise induced coherent oscillations…

A proposal for QED in half space is made. Starting from the well known principle of mirror charges in electrostatics, we formulate boundary conditions for electromagnetic fields and charge carrying currents both in the classical and the quantum context. Free classical and quantum fields are constructed, such that the required boundary conditions…

In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing…

We consider asymptotics of the correlation functions of characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$ and the hermitian sample covariance matrices $X_n=n^{-1}A_{m,n}^*A_{m,n}$. We use the integration over the Grassmann variables to obtain a convenient integral representation. Then we show that the asymptotics of…

We consider a passive scalar field under the action of pumping, diffusion and advection by a smooth flow with a Lagrangian chaos. We present theoretical arguments showing that scalar statistics is not conformal invariant and formulate new effective semi-analytic algorithm to model the scalar turbulence. We then carry massive numerics of passive…

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent \chi and the wandering exponent \xi are related through the universal relation \chi=2\xi -1, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. I will give a…

We discuss recently established criteria for the formation of extended states on tree graphs in the presence of disorder. These criteria have the surprising implication that for bounded random potentials, as in the Anderson model, in the weak disorder regime there is no transition to a spectral regime of Anderson localization in the form usually…

I shall describe a theory of adiabatic response for controlled open systems governed by Lindblad evolutions. The theory gives quantum response a geometric interpretation induced from the geometry of Hilbert space projections. For a two level system the metric turns out to be the Fubini-Study metric and the symplectic form the adiabatic curvature…

The fluctuations of a physical quantity can be described by its moments. In many cases, these diverge as an asymptotic parameter becomes large (or small), through the influence of geometric singularities. These large moments are described by power laws whose exponents can be determined from a knowledge of the singularities. Examples are twinkling…