# Seminars & Events for Mathematical Physics Seminar

##### Absence of mobility edge for the Anderson random potential on tree graphs at weak disorder

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 envisioned.

##### The universal relation between exponents in first-passage percolation

Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier's law for heat conduction from the laws of mechanics remains thus a major unsolved problem. In this note we present a deterministic mechanical model of a heat-conducting chain with nontrivial interactions, where kinetic energy fluctuations at the nodes of the chain are removed. In this model the derivation of Fourier's law can proceed rigorous

##### Fractal iso-contours of passive scalar in smooth random flows

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 scalar turbulence with the focus on the statistics of nodal lines. The distribution of contours over sizes and perimeters is shown to depend neither on the flow realization nor on the resolution (diffusion) scale $r_d$ for scales exceeding $r_d$. The scalar isolines are found fractal/smooth at the scales larger/smaller than the pumping scale $L$.

##### Characteristic polynomials of the hermitian Wigner and sample covariance matrices

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 the correlation functions of any even order coincide with that for the GUE up to a factor, depending only on the fourth moment of the common probability law of the matrix entries, i.e. that the higher moments do not contribute to the above asymptotics.

##### QED in Half Space

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 hold. Conservation laws are discussed. A variation of the principle of mirror charges is given, which leads to a dual set of boundary conditions and for which again free fields can be constructed.

##### Random natural frequencies, active dynamics and coherence stability in populations of coupled rotators

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. The talk will focus on the case on noisy dynamics, with different rotators stirred by independent Brownian motions. The approach we present is based on the observation that in the absence of disorder the Kuramoto model reduces to a Langevin dynamics for the mean field plane rotator (or classical XY spin) model. The analysis is carried at the level of the Fokker-Planck PDE for the evolution of the system's empirical density, in the limit where N tends to infinity.

##### Macdonald Processes and Some Applications in Probability and Integrable Systems

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-Laplace transforms. Taking limits and degenerations we arrive at new results in the study of certain directed polymers, branching processes, quantum many body systems, interacting particle systems and stochastic PDEs. This is based on joint work with Alexei Borodin.

##### Nonintersecting Random Walkers with a Staircase Initial Condition

We study a model of one dimensional particles, performing geometrically weighted random walks that are conditioned not to intersect. The walkers start at equidistant points and end at consecutive integers. A naturally associated tiling model can be viewed as one of placing boxes on a staircase. For a particular value of the parameters we obtain a known model for the Schur measure, which has the sine kernel as a scaling limit. However, for other parameter values the process at the local scale, close to the starting points, does not fall in the universality class of the sine kernel. Instead, as the number of walkers tends to infinity we obtain a new family of kernels describing the local correlations. We shall describe these limits and some of their interesting features. This is joint work with Maurice Duits.

##### Low Density Limit of BCS Theory and Bose-Einstein Condensation of Fermion Pairs

We consider the low density limit of a Fermi gas in the BCS approximation. We show that if the interaction potential allows for a two-particle bound state, the system at zero temperature is well approximated by the Gross-Pitaevskii functional, describing a Bose-Einstein condensate of fermion pairs. This is joint work with Robert Seiringer.

##### Invariant Measures and the Soliton Resolution Conjecture

The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. I will present a theorem that proves a "statistical version" of this conjecture at mass-subcritical nonlinearity. The proof involves a combination of techniques from large deviations, PDE, harmonic analysis and bare hands probability theory.

##### Diffusion of Wave Packets for the Markov Schroedinger Equation

The long time evolution of waves in a homogeneous random environment will be discussed. Proving that the wave amplitude evolves diffusively over any sufficiently long time scales remains an open problem. One obstacle that arises is recurrence -- return of portions of the wave packet to regions previously visited. However, if one removes recurrence by allowing the environment to evolve randomly in time, then diffusion of the wave amplitude can be proved in a relatively simple fashion.