This term, all seminars take place Thursdays at 2pm, room MS.04 (Zeeman Building), unless indicated otherwise.



Schedule for

Sabine Jansen, 12.06.2008

Seminars

14.01.2010 Jani Lukkarinen (Helsinki)
Kinetic Equations for a Homogeneous Bose Fluid with Condensate
We consider the kinetics of a three-dimensional fluid of weakly interacting bosons with supercritical densities. More precisely, we consider the postulated nonlinear Boltzmann-Nordheim equations for this system, in a spatially homogeneous state which has an isotropic momentum distribution. The resulting evolution equations have a surprisingly rich mathematical structure, where proper definitions play an important role. In fact, it is not even clear if the standard kinetic scheme for derivation of the Boltzmann-Nordheim equations can be trusted in the presence of a condensate. Elaborating on the work of Herbert Spohn [arXiv:0809.4551], we propose a definition of the coupled equations for which the thermal equilibrium states are stationary. To test the validity of the equations, we have studied global existence and uniqueness of solutions, as a problem about return to equilibrium from a perturbation of a thermal state with a condensate. I will discuss here preliminary results from this joint work with Jogia Bandyopadhyay and Antti Kupiainen.
21.01.2010 Michalis Loulakis (Crete)
Zero-range condensation at criticality
Zero-range processes with decreasing jump rates exhibit a continuous condensation transition, where a finite fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number after adding a subextensive excess mass of particles at the critical density. We establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from 0 to a positive value at a critical scale. Our results also include distributional limits for the fluctuations of the maximum in both regimes, which change from standard extreme value statistics to Gaussian. We identify the detailed behaviour at the critical scale including sub-leading terms, providing a full understanding of the crossover between the two regimes.
28.01.2010 Paul Chleboun (University of Warwick)
Finite size effects in a stochastic condensation model
We study finite size effects on the condensation transition in the zero-range process. The transition is already well understood in the thermodynamic limit, but finite size effects can be significantly large and counter intuitive. We observe a high current overshoot above the limiting critical value and an abrupt change between a putative fluid and condensed phase. This is reminiscent of a first order transition, although condensation is known to be continuous in the thermodynamic limit. Close to the abrupt transition we also observe metastable switching between the two phases. We can define an effective free energy landscape and thus predict the scaling of the lifetime of the two phases. We formulate approximations for the fluid and condensed phases, and use these to derive the leading order finite size effects.
04.02.2010 Thomas Østergaard Sørensen (Imperial College)
Regularity properties of Coulombic wavefunctions and their one-electron densities
We review recent results on the regularity and structure of wavefunctions psi of the non-relativistic Schrödinger operator describing atoms and molecules (that is, with Coulomb interactions). We also discuss the regularity of the associated one-electron densities rho. In particular, we characterize the structure of psi around 'two-particle coalescence points'. The method of proof of the latter extends to the study of the structure of the solutions to the multiconfiguration equations, and their densities, at the positions of the nuclei.
This is joint work with S. Fournais and M. and T. Hoffmann-Ostenhof.
11.02.2010 Amin Coja-Oghlan (University of Warwick)
A statistical mechanics perspective on hard computational problems
A large variety of computational problems can be classified as "computationally hard". In recent years researchers from statistical mechanics have investigated such problems via methods from the theory of spin glasses. The aim of this talk is to give a brief overview of this work, and of the extent to which these considerations can be turned into rigorous mathematics.
18.02.2010 Christina Goldschmidt (University of Warwick)
The scaling limit of critical random graphs
Consider the Erdös-Rényi random graph G(n,p): we have n vertices, every pair of which is joined by an edge independently with probability p. As is well known, this model undergoes a phase transition as p passes through 1/n. Below 1/n, there are only "small" components; above, there is one so-called giant component, which contains a positive proportion of the vertices. We are interested in the "critical window", where p = n-1 + λ n-4/3 for some real λ. In this regime, the largest components are of size n2/3 and have finite surpluses (where the surplus of a component is the number of edges more than a tree that it has). Using a bijective correspondence between graphs and certain "marked random walks", we are able to give a (surprisingly simple) metric space description of the scaling limit of the ordered sequence of components, where edges in the original graph are re-scaled by n-1/3. Along the way, I will talk about the notion of a continuum random tree; such trees form an important part of the description of a limiting component in the random graph. Our convergence holds in a sufficiently strong sense that we are able to deduce the convergence in distribution of the diameter of G(n,p), rescaled by n-1/3, to a non-degenerate random variable, for p in the critical window.
This is joint work with Louigi Addario-Berry (Université de Montréal) and Nicolas Broutin (INRIA Rocquencourt).
25.02.2010 Marek Biskup (UCLA)
Gibbs measures on permutations of the integers
I will discuss a problem that I learned from Daniel Ueltschi at this very location some 3 years ago. Consider a probability distribution on the set of all permutations of the integers that weighs a permutation by the exponential of the negative sum of the squares of the displacements between the integers and their images under the permutation. This problem arises as a caricature to Feynman's representation of interacting Bose gases. I will show how to formalize the above description in terms of infinite-volume Gibbs measures and then provide a full classification of all such measures by means of the quantity called a flow. In particular, all Gibbs measures are translation invariant and there is exactly one that has only finite cycles, almost surely. The talk is based on joint work -- and a paper under preparation -- with Thomas Richthammer (UCLA).
03.03.2010 Oliver Riordan (Oxford)
The Generalized Triangle-Triangle Transformation in Percolation
One of the main aims in the theory of percolation is to find the `critical probability' above which long range connections emerge from random local connections with a given pattern and certain individual probabilities. The quintessential example is Kesten's result from 1980 that if the edges of the square lattice are selected independently with probability p, then long range connections appear if and only if p>1/2. The starting point is a certain self-duality property, observed already in the early 60s; the difficulty is not in this observation, but in proving that self-duality does imply criticality in this setting.
Since Kesten's result, more complicated duality properties have been used to determine a variety of other critical probabilities. Recently, Scullard and Ziff have described a very general class of self-dual planar percolation models; we show that for the entire class (in fact, a larger class), self-duality does imply criticality.
Joint seminar with DIMAP, takes place Wednesday in B3.02 at 2pm.
04.03.2010 Hermann Schulz-Baldes (University of Erlangen)
Semicircle law and linear statistics for a class of random matrices with correlated entries
For random matrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner's argument showing the convergence of the density of states to the semicircle law. Joint work with J. Schenker.
11.03.2010

18.03.2010 Hillel Raz (University of Cardiff)
Locality Bounds in the Anharmonic Lattice
Local operators are defined as operators which act on local finite sets and outside of that set act trivially. In this talk we seek to answer the following question, what happens to local operators evolved according to some Hamiltonian. That is, we seek to find whether or not such operators remain local and if so to what extent. We do this by comparing the evolved local operator with another local operator that is not evolved and bounding a relation between the two of them. The result is known as the Lieb-Robinson bounds, a type of locality bounds. We provide such a bound in a few specific settings, in particular in the anharmonic lattice.