Thursday 12 November (room D1.07)

09:30-10:20  •  Dima Ioffe  •  Finite connections for the super-critical Bernoulli bond percolation in 2D

We derive sharp asymptotics of finite connections (connections via finite clusters) for the super-critical Bernoulli bond percolation in 2D. A well known analog of the resulting formula holds for the truncated two point function in the context of the exactly soluable 2D low temperature Ising model. Our approach relies on recent advances in the Ornstein-Zernike theory which is built upon a stochastic geometric representation of long sub-critical clusters. The self-duality of the model plays, therefore, an essential role - the main contribution to the asymptotics comes from long open dual loop which encircle two distant directly connected points. (Joint work with Massimo Campanino and Oren Louidor.)

10:20-10:50  •  coffee break

10:50-11:40  •  Jean-Dominique Deuschel  •  Hydrodynamical Limit for the Landau Ginsburg model with non-convex potential

11:40-12:30  •  Peter Mörters  •  Random networks with nonlinear preferential attachment

We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a nonlinear function of their degree. Our main interest is in the phase transitions occuring when we vary the attachment function and move from weak to strong preferential attachment. Properties discussed include the degree distribution, existence of a hub and clustering. This is joint work with Steffen Dereich (Marburg).

12:30-14:00  •  lunch

14:00-14:50  •  Christian Borgs  •  Pólya Urns and Convergence of Preferential Attachment Graphs

After discussing general notions of graph sequences, including the Benjamini-Schramm limit of a sequence of graphs, I introduce a new representation for preferential attachments graphs related to an old concept from probability, the notion of Pólya urns. Using this new representation, we then construct the Benjamini-Schramm limit of a sequence of preferential attachment graphs. The work presented in this talk is joint work with N. Berger, J.T. Chayes and A. Saberi.

14:50-15:40  •  Wilfrid Kendall  •  The Poissonian City

I will talk on some work following up on Aldous and Kendall (Short-length routes in low-cost networks via Poisson line patterns. Adv. in Appl. Probab. 40  (2008), no. 1, 1-21). In the previous work we showed how Poisson line processes could be used to deliver a striking resolution of a frustrated optimization problem in network theory. The follow-up work includes: the use of Lévy subordinators to study random variation of the lengths of approximate geodesics; establishing rather strong lower bounds by using classical inequalities for Mills ratios; and the introduction of a curious improper anisotropic line pattern in order to get information about flows in a Poissonian line network (a "Poissonian city"). If time permits, I'll also talk about some questions to which I do not know the answer...

15:40-16:00  •  tea break

16:00-17:00  •  Departmental Colloquium (Michel Crucifix, Reduced order models of palaeoclimates : techniques, challenges and promises, B3.02)

17:00-18:30  •  Italian buffet & wines (in math common room)

18:30  •  Round table about the future of statistical mechanics

09:30-10:20  •  Roberto Fernández  •  Convergence of cluster expansions: Kirkwood-Salzburg equations versus inductive Kotecký-Preiss-Dobrushin arguments

I will compare two approaches to establish convergence regions of cluster expansions of random geometric objects: (i) Through solutions of Kirkwood-Salzburg equations and (ii) through inductive arguments pioneered by Kotecký and Preiss and refined by Dobrushin.  In recent work with Rodrigo Bissacot and Aldo Procacci (both from the University of Minas Gerais, Brazil) we have shown: (a) Both approaches yield the same open convergence region, but the inductive approach extends convergence to its boundary. (b) The corresponding criterion is the same initially obtained by Gruber and Kunz (for the open region). (c) The Kirkwood-Salzburg argument admits a simple modification that leads to convergence also in the boundary of the region.

10:20-10:50  •  coffee break

10:50-11:40  •  Stephan Luckhaus  •  Lattice based Hamiltonians, two scale convergence and nonlinear elasticity

We are looking at lattice based Hamiltonians modelling elastic interactions, under growth assumptions. We prove a large deviations result with an energy functional of nonlinear elastic type as the rate functional. Local fluctuations are described by a parametrized gradient Gibbs (Young) measure. (Joint work with Roman Kotecký.)

11:40-12:30  •  Balint Tóth  •  Erdös-Rényi random graphs + forest fires = self-organized criticality

12:30-14:30  •  lunch

14:30-15:20  •  John Cardy  •  Classical and quantum localization in two and three dimensions

Consider a closed directed graph G in which each node has exactly two incoming and two outgoing edges. In the classical problem, the two ways of decomposing each node into two disjoint directed pieces are assigned probabilities p and 1-p. This decomposes G into a union of closed loops. A particle moving on G therefore performs a deterministic walk in a random medium. The quantum problem is similar, except that transition probabilities at the nodes are replaced by amplitudes, and the particle's wave-function can rotate as it propagates along each edge. It turns out that there is an exact mapping between a particular class of quantum problems with quenched edge disorder and the classical problem. An interesting question when G is a regular lattice embedded in R^d is whether, in the limit of an infinite lattice, almost all loops have finite length (corresponding to all states being localized), or whether there is a finite probability of escape (corresponding to extended states.) We consider this on some lattices in d=2 and 3 where the classical problem can be related to percolation, and argue that for d=2 almost all states tend to be localized while for d=3 there is a transition between localized and extended states as a function of p.

15:20-15:50  •  tea break

15:50-16:40  •  David Brydges  •  Self-avoiding Walk and the Renormalisation Group

Self-avoiding walks on Z^d are simple-random walk paths without self-interstections. Self-avoiding walks of the same length are declared to be equally likely. The basic question is how far on average is their endpoint from the origin? After a brief review of the current state of knowledge I will describe work in progress with Gordon Slade for the case d = 4 which is an application of the renormalisation group to a supersymmetric lattice field theory. Our immediate goal is to prove that the critical two-point function (Green function) for a spread-out model of self-avoiding walks on Z^d decays like |x|^{-2} at large distances, as it does for simple random walk.

16:40-17:30  •  Alan Hammond  •  Phase boundary fluctuations and maximum local roughness

Condition supercritical percolation on the squre lattice so that the origin is enclosed by a dual circuit whose interior traps an area of n^2. The Wulff problem concerns the shape of the circuit. We study the circuit's fluctuation. A well-known measure of this fluctuation is maximum local roughness (MLR), which is the greatest distance from a point on the circuit to the boundary of circuit's convex hull. Another is maximum facet length (MFL), the length of the longest line segment of which this convex hull is comprised. In a forthcoming article, I prove that, for various models including supercritical percolation, under the conditioned measure, MLR = \Theta(n^{1/3}(\log n)^{2/3}) and MFL = \Theta(n^{2/3}(log n)^{1/3}). The essential hypotheses that are needed for these results are translation invariance, exponential decay of dual connectivity, bounded energy and the ratio-weak-mixing assumption (which I'll discuss if there is enough time). An important tool is a result establishing the profusion of regeneration sites in the circuit. The talk will focus on deriving the main results with this tool.

19:00  •  Conference dinner

09:30-10:20  •  Wolfgang König  •  Phase Transitions for Dilute Particle Systems with Lennard-Jones Potential

We consider a classical dilute particle system in a large container with pair-interaction given by a Lennard-Jones-type potential. The inverse temperature is picked proportionally to the logarithm of the number of particles. We identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity. Loosely speaking, the lower the temperature, the larger the relevant crystal structures that give the main contribution to the free energy. The phases are characterised by the size of the relevant configurations. Our main tool is a new large deviation principle for sparse point configurations. (Joint work with A. Collevecchio, P. Mörters, and N. Sidorova.)

10:20-10:50  •  coffee break

10:50-11:40  •  Ostap Hryniv  •  Phase Transition for a Model of Microtubule Growth

We study a continuous time stochastic process on strings made of two types of particles, whose dynamics mimics the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterisation of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

11:40-12:30  •  Nilanjana Datta  •  Min- and Max- Relative Entropies

Two new relative entropy quantities are introduced and their properties, including their relation with the quantum relative entropy, are investigated. The operational significance of these quantities, in the context of Quantum Information Theory, is discussed. Further, these relative entropies are seen to lead naturally to the definition of two entanglement monotones, whose properties are also discussed. (Previous knowledge on Quantum Information will not be assumed)