13rd Itzykson Conference PUZZLES OF GROWTH


The purpose of the 13th Itzykson meeting "PUZZLES OF GROWTH" is to present the recent advances in the study of growth phenomena, putting together mathematicians and physicists in a relaxed ambiance.

It will take place in Saclay at the Institut de Physique Théorique (formerly known as Service de Physique Théorique), Orme des Merisiers, CEA-Saclay, from June 9 to June 11, 2008.

It is preceded by the Enrage Topical School ON GROWTH AND SHAPES at Institut Henri Poincaré, Paris.


E. Ben-Jacob,Tel Aviv University
F. Camia, Vrije Universiteit Amsterdam
S. Dorogovtsev, Universidade de Aveiro
M. Drmota, Technische Unversität Wien
F. David, IPhT, Saclay
B. Duplantier, IPhT, Saclay
E. Guitter, IPhT, Saclay
W. Janke, Universität Leipzig
T. Kennedy, University of Arizona
K. Kytölä, Université Paris Sud, Orsay
A. Middleton, Syracuse University
A. Okounkov, Princeton Unversity
S. Redner, Boston University
H. Saleur, IPhT, Saclay
R. Santachiara, Ecole Normale Supérieure, Paris
S. Smirnov, Université de Genève
W. Werner, Université Paris Sud, Orsay


Monday, June 9

 8.15 Bus at the RER B station "Orsay-Le Guichet"

 8.45 Registration
 9.15 Welcome word

 9.30 Wendelin Werner: Are frontiers symmetric?
10.25 Bertrand Duplantier: Quantum Gravity and Brownian Large Deviations
11.20 Coffee break
11.50 Eshel Ben Jacob: The Mathematical Skills of Bacteria

12.45 Lunch
14.15 Kalle Kytölä: Some CFT fusions from SLE local martingales
15.10 Alan Middleton: Exploring the effects of disorder on geometry
16.05 Coffee break
16.35 François David: TBA

18.00 Bus to RER B

Tuesday, June 10

 9.00 Bus at the RER B station "Orsay-Le Guichet"

 9.30 Raoul Santachiara: Interfaces in lattice Z(N) models
10.25 Wolfhard Janke: Percolating Excitations - A Geometrical View of Phase Transitions
11.20 Coffee break
11.50 Michael Drmota: Large Random Planar Graphs

12.45 Lunch

14.15 Tom Kennedy: Testing for SLE using the driving process
15.10 Stanislas Smirnov: Ising lattice universality
16.05 Coffee break
16.35 Hubert Saleur: Boundary loop models

18.00 Bus to RER B

Wednesday, June 11

 9.00 Bus at the RER B station "Orsay-Le Guichet"

 9.30 Andrei Okounkov: Noncommutative geometry of planar dimers
10.25 Federico Camia: Scaling Limits of 2D Percolation
11.20 Coffee break
11.50 Sergei Dorogovtsev: Transition from finite to infinite-dimensional trees

12.45 Lunch

14.15 Sidney Redner: Cutting Corners
15.10 Emmanuel Guitter: The three-point function of planar quadrangulations

16.05 Closing word

16.45 Bus to RER B

Program of the talks

Federico Camia, Amsterdam University Scaling Limits of 2D Percolation I will discuss some of the recent progress in the study of the critical and off-critical percolation scaling limits in two dimensions. In particular, I will focus on the scaling limit of collections of cluster boundaries and its connection to the Conformal Loop Ensembles (CLEs) introduced by Sheffield and Werner. S. N. Dorogovtsev, University of Aveiro and Ioffe Institute, St. Petersburg Transition from finite- to infinite-dimensional trees Unlike non-exotic equilibrium random trees, which are finite-dimensional, growing trees can have finite or infinite Hausdorff dimensions depending on a model. We discuss a sharp transition between these two contrasting regimes in generalized random recursive trees and compare it with similar phenomena in equilibrium loopy systems: long-range percolation and related problems.
[1] S. N. Dorogovtsev, P. L. Krapivsky, and J. F. F. Mendes, Transition from small to large world in growing networks, EPL 81, 30004 (2008).
[2] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. (2008); arXiv:0705.0010.
[3] S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin, and A. Y. Zyuzin, Organization of modular networks, arXiv:0803.3422
Michael Drmota, TU Wien Large Random Planar Graphs The study of planar graphs has a long history. Nevertheless the study of random planar graphs has started only few years ago by A. Denise, M. Vasconcellos and D. J. A. Welsh in 1996. Since then much attention has been payed to this topic. A ground-breaking result was obtained recently by O. Gimenez and M. Noy by solving the long-standing open problem of getting a precise estimates for the number of planar graphs, drawing on previous work by Bender et al. The purpose of this talk it present first a survey on asymptotic properties of random planar graphs. We will then focus on an expicit representation of the asymptotic degree distribution, a result that has been obtained jointly with Gimenez and Noy: the probability that a random node in a large random planar graph has degree k converges to p(k), where the generating function of the numbers p(k) can be explicitly stated and, as k to infinity, one has p(k) ~ c k1/2 Rk (for some constants c>0 and 0< R<1). Bertrand Duplantier, IPhT Saclay Quantum Gravity and Brownian Large Deviations The KPZ relation (Knizhnik, Polyakov, Zamolodchikov, 1988) of two-dimensional quantum gravity relates critical exponents in the Euclidean plane to those in presence of critical fluctuations of the metric. A rigorous proof of the KPZ relation is given for the Liouville field theory, in terms of the large deviations properties of the two-dimensional Gaussian free field and its associated Brownian motions (joint work with Scott Sheffield, Courant Institute). Emmanuel Guitter, IPhT Saclay The three-point function of planar quadrangulations I will give a derivation of the generating function for random planar quadrangulations with three marked vertices at prescribed pairwise distances. This derivation is based on a new bijection by Miermont between quadrangulations with marked vertices and delays, and so-called well-labeled maps. I will discuss the (universal) scaling limit of large quadrangulations, as well as various limiting regimes, when some of the distances become large or small. Eshel Ben Jacob, Tel Aviv University The Mathematical Skills of Bacteria Under natural growth conditions, bacteria can utilize intricate communication capabilities(e.g. quorum-sensing, chemotactic signaling and plasmid exchange) to cooperatively form (self-organize) complex colonies with elevated adaptability -- the colonial pattern is collectively engineered according to the encountered environmental conditions. Bacteria do not genetically store all the information required for creating all possible patterns. Instead, additional information is cooperatively generated as required for the colonial self-organization to proceed. We describe how complex colonial forms (patterns), emerge through the communication-based singular interplay between individual bacteria and the colony. Each bacterium is, by itself, a biotic autonomous system with its own internal cellular informatics capabilities (storage, processing and assessment of information). These afford the cell plasticity to select its response to biochemical messages it receives, including self-alteration and the broadcasting of messages to initiate alterations in other bacteria. Hence, new features can collectively emerge during self-organization from the intracellular level to the whole colony. The cells thus assume newly co-generated traits and abilities that are not explicitly stored in the genetic information of the individuals. Wolfhard Janke, Leipzig University Percolating Excitations - A Geometrical View of Phase Transitions Many spin and lattice gauge models admit an alternative description in terms of geometrical objects. In the talk I will discuss to which extent suitably defined geometrical excitation networks may encode in their percolation properties and fractal structure thermal critical behaviour. Examples include the two-dimensional Potts model for which two types of spin clusters can be defined. Whereas Fortuin-Kasteleyn clusters are related to the standard critical behaviour of the pure model, geometrical clusters describe the tricritical behaviour that arises when including vacant sites in the pure Potts model. Another class of models are O(n) spin models in three dimensions where high-temperature graphs yield such a geometrical view of the standard phase transitions. In all cases the geometrical picture is supported by Monte Carlo simulations. In an outlook, further possible applications of the geometrical viewpoint to other systems are briefly discussed. Tom Kennedy, University of Arizona Testing for SLE using the driving process One way to test if a model of random curves in the plane is SLE is to compute the stochastic driving process that describes the curves through the Loewner equation and see if this process is a Brownian motion. We simulate several models (some of which are SLE and some of which are not), numerically compute their driving process, and then test if it is a Brownian motions. Our goal is to see how well one can determine whether or not a model is SLE by studying this stochastic driving process and to compare various tests that the driving process is a Brownian motion. We also describe an implementation of the zipper algorithm for numerically computing the driving function which runs in a time O(N^1.35) rather than the usual O(N^2), where N is the number of points on the curve. Kalle Kytölä, LPTMS Orsay Some CFT fusions from SLE local martingales Schramm-Loewner Evolutions (SLE) are growth processes that describe continuum limits of interfaces in 2-d statistical physics at criticality. Appropriate spaces of local martingales of the processes carry a representation of the Virasoro algebra. We will discuss in light of examples how this space is related to the fusion product of the boundary condition changing fields. The first case shows that an absolute vacuum component of certain fusion contains information about the question of chordal SLE reversibility. The other examples specialize to fusion producs in critical percolation. Here we will in particular see the appearance of logarithmic Virasoro modules. Alan Middleton, Syracuse University Exploring the effects of disorder on geometry Real systems, ranging from materials such as spin glasses through interfaces in porous medium through networks of roads, have pervasive disorder. Geometrical features of these models, such as domain walls or shortest paths, can be described by nontrivial scaling. I will focus on numerical results that support analytical descriptions of these models, such as in driven interfaces, and suggest startling symmetries, such as the possibility of SLE in 2D spin glasses. In addition, optimization algorithms are proposed as a method for studying coarsening of domain walls in disordered models. Andrei Okounkov, IAS Princeton Noncommutative geometry of planar dimers   Sid Redner, Boston University Cutting Corners We discuss two simple models for shrinking: (i) the erosion of a rock by chipping exposed corners and (ii) the smoothing of corners in the kinetic Ising model. In the rock erosion model, each chip is small so that only a single corner and a fraction of its two adjacent sides are cut away in a single chipping event. After many chipping events, the rock is not round, facet lengths and corner angles distributed over a broad range, and there are large fluctuations between realizations. In the Ising model, we investigate the evolution of a single interface between ordered phases in two dimensions with either one corner or two corners. In both examples, the interface evolves to a limiting self-similar form. For the single corner system, we discuss a correspondence between the interface and the Young diagram that represents the partition of the integers. Hubert Saleur, IPhT Saclay Boundary loop models 2D Loop models have been shown recently to admit infinite families of different conformal boundary conditions. In this talk, I will review these boundary conditions and the corresponding critical exponents. I will briefly describe the underlying boundary Coulomb gas and boundary Temperley Lieb algebras formalism. I will finally discuss some combinatorial applications. Raoul Santachiara, LPT ENS Paris Interfaces in lattice Z(N) models We will discuss a family of critical lattice spin models which includes the Ising and the three-states Potts model. These models are described in the continuum limit by conformal field theories with additional Z(N) symmetries, the so called parafermionic theories. In particular we will present analitical and numerical results on interfaces which are good SLE candidates. The problem of the connection between SLE and extended conformal field theories will then be adressed. Stanislas Smirnov, Geneva University Ising lattice universality Wendelin Werner, ENS Paris Are frontiers symmetric?


Michel Bauer, IPhT CEA-Saclay
Denis Bernard, LPTENS Ecole Normale Supérieure, Paris
Zdzislaw Burda, Uniwersytet Jagiellonski, Krakow
François David, IPhT CEA-Saclay
Alexandre Lefèvre, IPhT CEA-Saclay


Participation in the school is open subject only to space constraints. There is no registration fee. Registration is mandatory, even for Paris-area scientists. Deadline for registration: May 2, 2008. It is still possible to register.


The list of participant is available online here.


We do not organize lodging for these two events, but you will find on our accommodation page many useful informations for your stay in Paris & in Saclay.


The conference is organised by the Institut de Physique Théorique - CEA Saclay. It is supported by the Commissariat à l'Energie Atomique (Direction des Sciences de la Matière), by the Agence Nationale de la Recherche (ANR) and by the Marie Curie Research and Training Network ENRAGE (European Network on RAndom GEometry MRTN-CT-2004-005616)


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