MATHEMATICAL PHYSICS SEMINAR

 Department of Mathematics
Princeton University
Princeton University Department of Mathematics Seminars Spring 2011 Schedule

FALL 2011 Lectures

Regular meeting time: Tuesdays 4:30-5:30
Place: Jadwin A06

**Please note location**

Date Speaker Title
Oct. 11, 4:30 p.m.
**A06 Jadwin Hall**		 Simone Warzel, TU- Munich 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
Oct. 18, 4:30 p.m.
**IAS - Room S-101 (Simonyi Hall)**		 Sourav Chatterjee, Courant Inst. NYU

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 rigorously.
Oct. 25, 4:30 p.m.
**A06 Jadwin Hall**
 Marija Vucelja, Courant Inst., NYU 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 $L$. We characterize the statistics of bending of a long isoline by the driving function of the L\"owner map, show that it behaves like diffusion with the diffusivity independent of resolution yet, most surprisingly, dependent on the velocity realization and the time of scalar evoluti! on.
Nov. 1, 4:30 p.m.
**IAS - Room S-101 (Simonyi Hall)**		 Tatyana Shcherbina, Institute for Low Temperature Physics, Kharkov, Ukraine 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.
Nov. 8, 4:30 p.m.
**A06 Jadwin Hall**
Michael Damron, Princeton University A simplified proof of the relation between scaling exponents in first-passage percolation
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 paths between the same points. This relation is sometimes referred to as the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution.
Nov. 15, 4:30 p.m.
**IAS - Room S-101 (Simonyi Hall)**

 Robert Schrader,
FU - Berlin 		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.
Nov. 22, 4:30 p.m.
**A06 Jadwin Hall** 		Giambattista Giacomin, Université Paris Diderot 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.
Nov. 29, 4:30 p.m.
**IAS - Room S-101** 		Ivan Corwin,
Courant Inst. NYU		 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.

 

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