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.
Speakers
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
Schedule
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 UniversityScaling Limits of 2D PercolationI 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. PetersburgTransition 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:0803.3422Michael Drmota, TU WienLarge 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 SaclayQuantum 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 SaclayThe 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 UniversityThe 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 UniversityPercolating 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 ArizonaTesting 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 OrsaySome 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 UniversityExploring the effects of disorder on geometryReal 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 PrincetonNoncommutative geometry of planar dimersSid Redner, Boston UniversityCutting 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 SaclayBoundary 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 ParisInterfaces 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 UniversityIsing lattice universalityWendelin Werner, ENS ParisAre frontiers symmetric?
Organisation
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
Registration
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.
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.
