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Presentation of the conference
This conference on recent progress in the research on random media is organized within the ANR project MEMEMO II (Random walks, random media and random models with reinforcement).
The main focus will be given to lectures of invited speakers, together with talks by young researchers.
The conference takes place on June 2-6, 2014 in the CIRM (International center for mathematical meetings) in Luminy, nearby Marseilles.
Inscriptions are managed through the conference page on the CIRM website.
Scientific comitee:
- Fabienne Castell (Université d'Aix-Marseille)
- Jean-Marc Derrien (Université de Bretagne Occidentale)
- Nathanaël Enriquez (Université Paris 10)
- Nadine Guillotin-Plantard (Université Lyon 1)
- Yueyun Hu (Université Paris 13)
- Zhan Shi (Université Paris 6)
Organizers:
- Nathanaël Enriquez (Université Paris 10)
- Yueyun Hu (Université Paris 13)
- Zhan Shi (Université Paris 6)
Contacts: Nathanaël Enriquez, Yueyun Hu, Zhan Shi
Invited speakers
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- Louigi Addario-Berry (Université Mc Gill) :
Growing random trees, maps, and squarings — Abstract — Slides
We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. The sequence of maps has an almost sure limit G; we show that G is the distributional local limit of large, uniformly random 3-connected graphs.
A classical result of Brooks, Smith, Stone and Tutte associates squarings of rectangles to edge-rooted planar graphs. Our map growth procedure induces a growing sequence of squarings, which we show has an almost sure limit: an infinite squaring of a finite rectangle, which almost surely has a unique point of accumulation. We know almost nothing about the limit, but it should be in some way related to "Liouville quantum gravity".
Parts of this are based on joint work with Nicholas Leavitt.
- Gérard Ben Arous (Courant Institute) :
Scaling limits for slow random walks
- Frank den Hollander (Leiden University) :
Annealed scaling for a charged polymer — Abstract — Slides
We study an undirected polymer chain living on the one-dimensional
integer lattice and carrying i.i.d. random charges. Each self-intersection
of the polymer chain contributes an energy to the interaction Hamiltonian
that is equal to the product of the charges of the two monomers that meet.
The joint probability distribution for the polymer chain and the charges
is given by the Gibbs distribution associated with the interaction
Hamiltonian. We analyze the annealed free energy per monomer in the
limit as the length of the polymer chain tends to infinity.
We derive a spectral representation for the free energy and use this
to show that there is a critical curve in the (charge bias, inverse
temperature)-plane separating a ballistic phase from a subballistic
phase. We show that the phase transition is first order, identify the
scaling behaviour of the critical curve for small and for large charge
bias, and also identify the scaling behaviour of the free energy for
small charge bias and small inverse temperature. In addition, we prove
a large deviation principle for the joint law of the empirical speed
and the empirical charge, and derive a spectral representation for the
associated rate function. This in turn leads to a law of large numbers
and a central limit theorem.
What happens for the quenched free energy per monomer remains open. We
state two modest results and raise a few questions.
Joint work with F. Caravenna, N. Petrelis and J. Poisat
- Gady Kozma (Weizmann Institute) :
Noncommutative lace expansion — Abstract
Lace expansion is a method for solving self-interacting models in high dimension. In the core of the technique is the problem of solving convolution equations. Initially highly dependent upon Fourier transform, in the last decade it has evolved to the degree that it can be applied in some non-commutative settings where Fourier transform is not available.
Joint work with Remco van der Hofstad and Erwin Bolthausen.
- Lionel Levine (Cornell University) :
Scaling limit of the abelian sandpile on Z2 — Abstract
—
Slides
Which functions of two real variables can be expressed as limits of
superharmonic functions from (1/n)Z2 to (1/n2)Z? I'll discuss joint
work with Wesley Pegden and Charles Smart on the case of quadratic
functions, where this question has a surprising and beautiful answer:
the maximal such quadratics are classified by the circles in a certain
Apollonian circle packing of the plane. I'll also explain where the
question came from (the title is a hint!).
Sandpiles and system-spanning avalanches — Abstract
A sandpile on a graph is an integer-valued function on the vertices.
It evolves according to local moves called topplings. Some sandpiles
stabilize after a finite number of topplings, while others topple
forever. For any sandpile s0 if we repeatedly add a grain of sand at
an independent random vertex, we eventually reach a sandpile sτ
that topples forever. Statistical physicists Poghosyan, Poghosyan,
Priezzhev and Ruelle conjectured a precise value for the expected
amount of sand in sτ in the limit as s0 goes to negative
infinity. I will outline the proof of this conjecture in
arXiv:1402.3283 and explain the big-picture motivation,
which is to give more predictive power to the theory of
"self-organized criticality".
- Satya Majumdar (Université Paris 11) :
Top Eigenvalue of a Random Matrix: A tale of tails — Abstract — Slides
The statistical properties of the largest eigenvalue of a random matrix
are of interest in diverse fields such as in the stability of large
ecosystems, in disordered systems, in statistical data analysis and even
in string theory. In this talk I'll discuss some recent developments in
the theory of extremely rare fluctuations (large deviations) of the
largest eigenvalue using a Coulomb gas method. Such rare fluctuations have
also been measured in recent experiments in coupled laser systems. I'll
also discuss recent applications of this Coulomb gas method in three
different problems: entanglement in a bipartite system, conductance
fluctuation through a mesoscopic cavity and the vicious random walkers
problem.
Random Convex Hulls: Applications to Ecology and Animal Epidemics — Slides
- Stanislav Volkov (Lund University) :
Les subdivisions géométriques stochastiques — Abstract — Slides
Je présenterai plusieurs modèles de subdivisions géométriques aléatoires, de type de Diaconis et Miclo (2011), où un triangle se subdivise en six petits triangles par ses médianes et se remplace par l'un de ces triangles au hasard. Le processus se poursuit à l'infini. Je vais montrer que dans un modèle similaire la forme limite qui arrive de subdivision d'un quadrilatère est un parallélogramme. Je vais aussi montrer que subdivisions géométriques d'un triangle par les bissectrices convergent à une distribution non-atomique. Finalement, je vais expliquer pourquoi les subdivisions d'un triangle en choisissant un point au hasard sur chaque côté converge vers un triangle "plat", de manière similaire au résultat de Diaconis et Miclo.
L'urne harmonique simple — Abstract — Slides
L'urne harmonique simple est un processus stochastique à temps discret correspondant au portrait de phase de l'oscillateur harmonique simple. Cette urne est une version d'urne de Pólya à deux couleurs généralisée avec renforcement négatif-positif, et dans un sens, elle peut se considérer comme un type de « mariage » entre l'urne de Friedman et le modèle fusillade d'O.K. Corral, où nous redémarrons le processus chaque fois que l’on manque de boules de certaine couleur en commutant les couleurs des boules. Nous montrons la transience du processus en utilisant différents couplages avec les processus de naissance et de décès et les processus de renouvellement. Il s'avère que l'urne harmonique simple est vraiment à peine transiente, parce qu'une modification mineure du modèle la fait récurrente.
Travail en commun avec Edward Crane, Nicolas Georgiou, Richard Waters et Andrew Wade.
- Ofer Zeitouni (Weizmann Institute and University of Minnesota) :
Branching structures, extremal processes, thick points, and Liouville Brownian motion:
Freezing and extremal processes — Abstract
The freezing in the title refers to a property of point processes: let (Xi)i≥1 denote a point process which is locally finite and has finite maximum. For a function f continuous of compact support, define Zf=f(X1)+f(X2)+.... We say that freezing occurs if the Laplace transform of Zf depends on f only through a shift. I will discuss this notion and its equivalence with other properties of the point process. In particular, such freezing occurs for the extremal process in branching random walks and in certain versions of the (discrete) two dimensional GFF.
Joint work with Eliran Subag
Thick points of the GFF and heat kernel estimates for Liouville Brownian motion — Abstract
The structure of thick points of the GFF dominate the structure of the (subcritical) multiplicative chaos
constructed from it. We will discuss how they also control heat kernel estimates for Liouville Brownian motion (obtained by time changing a planar Brownian motion in function of a subcritical Gaussian multiplicative chaos).
Joint work with Pascal Maillard, Remi Rhodes, Vincent Vargas
Schedule
Schedule of the week.
List of talks
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Participants
An up-to-date version is available on the CIRM website.
Practical matters
Inscriptions are handled via the conference page on the CIRM website.
Informations about getting to the CIRM can be found by clicking here.
Links
Previous meetings of the ANR project Mememo2:
and of the ANR Mememo:
- 10 - 14 mai 2010 in Luminy (CIRM) : Page web
- 8 - 12 juin 2009 in Aussois : Page web
- 26 - 30 mai 2008 in Plougonvelin : Page web
