#### Mini-courses

##### ⊗ Gerard Ben Arous (Courant Institute)

###### Slow dynamics in random media

Complex systems tend to reach equilibrium slowly. This course describes a general and universal mechanism for these slow relaxation phenomena and its consequences as aging and anomalous diffusion. This mechanism will be illustrated by questions about dynamics of spin-glasses and about random walks on percolation clusters.

- Lecture 1: The Bouchaud trap model and its scaling limits.
- Lecture 2: The general trapping mechanism in dimension 1: Randomly Trapped Brownian Motions
- Lecture 3: Trapping and aging for random walks on critical and supercritical trees
- Lecture 4: Universality for energy landscapes of mean-field spin glasses
- Lecture 5: Universality for dynamics of mean-field spin-glasses

##### ⊗ Dmitry Ioffe (Technion, Haifa)

###### Stochastic geometry of classical and quantum Ising models

- Lectures 1-3: Path integral approach to Ising spin systems.
*FK*(Fortuin-Kesteleyn) and*RC*(random current) representations of classical Ising models. Switching lemma.*Example*: Exponential decay of correlations at non-zero magnetic fields.*FK*and*RC*representations for quantum Ising models in a transverse field. - Lecture 4: Quantum version of Erdös-Rényi random graphs.
- Lecture 5: Quantum Curie-Weiss model in a transverse field.

**References**

- M. Aizenman,
*Geometric analysis of*, Comm. Math. Phys.*φ*fields and Ising models. I, II^{4}**86**(1), 1-48 (1982). - M. Aizenman, A. Klein, , C. M. Newman,
*Percolation methods for disordered quantum Ising models*, Mathematics, Physics, Biology, R. Kotecky ed., 1-24, World Scientific, Singapore (1993). - M. Aizenman, B. Nachtergaele,
*Geometric aspects of quantum spin states.*Comm. Math. Phys.**164**(1), 17-63 (1994). - L. Chayes, N. Crawford, D. Ioffe, A. Levit,
*The Phase Diagram of the Quantum Curie-Weiss Model*, available at*arxiv:0804.1605*. - B. Bollobás, G. Grimmett, S. Janson,
*The random cluster model on the complete graph*Prob. Theory Rel. Fields**104**(3), 283-317 (1996). - M. Campanino, A. Klein, J. F. Perez,
*Localization in the ground state of the Ising model with a random transverse field*, Comm. Math. Phys.**135**(3), 499-515 (1991). - D. Ioffe, A. Levit,
*Long range order and giant components of quantum random graphs*, Mark. Proc. Rel. Fields**13**(3), 469-492 (2007). - S. Janson, T. Luczak, A. Ruci,
*Random Graphs*, John Wiley and Sons (2000).

##### ⊗ D. Chelkak (St. Petersburg) & S. Smirnov (Geneva)

###### SLE and conformal invariance for critical Ising model

- Introduction
- Discrete holomorphic and discrete harmonic functions
- Holomorphic observables in the Ising model
*SLE*and the interfaces in the Ising model- Further developments

- S. Smirnov,
*Towards conformal invariance of 2D lattice models*, International Congress of Mathematicians. Vol. II, Europ. Math. Soc., Zürich, 2006, 1421--1451 (available at*arXiv:0708.0032*). - S. Smirnov,
*Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model*, available at*arXiv:0708.0039*.

#### Conference talks

##### ⊗ Lorenzo Bertini (Università di Roma La Sapienza)

###### Small noise asymptotic for the Gallavotti-Cohen functional in diffusion processes

Consider a diffusion process in
* R^{n}*. It has been
shown that a suitable functional on the path space, interpreted as the
production of Gibbs entropy, has a large deviation rate function, as time
diverges, which satisfies the fluctuation theorem of Gallavotti and
Cohen. For a particular class of non reversible diffusions, I will
consider the small noise asymptotic of such rate function and show
that it can be expressed in terms of a suitable variational
problem. Some simple examples in which this limiting functional can
be written in a closed form are discussed.

##### ⊗ Federico Camia (Vrije Universiteit Amsterdam)

###### Large-*N* behavior of Mandelbrot's fractal percolation

Mandelbrot's fractal percolation process generates random fractal
sets by an iterative procedure which starts by dividing the unit cube
*[0,1] ^{d}* in

*equal subcubes, independently retaining each subcube with probability*

**N**^{d}*p*, and discarding the rest. This step is then repeated within the retained subcubes at all scales. As

*p*is varied, there is a percolation phase transition in terms of

*paths*for all

*d>1*, and in terms of

*(d-1)*-dimensional

*sheets*for all

*d>2*.

After introducing the model and some known results, I will present new results
concerning the discontinuity of crossing probabilities for all
*d>1*, and the large-*N* behavior of critical
values in dimension *2*.

(Joint work with Erik Broman)

##### ⊗ Francesco Guerra (Università di Roma La Sapienza)

###### Mean field spin glasses and diluted ferromagnets

We give a short review of the comparison arguments and the interpolation methods exploited in the rigorous study of mean field spin glass theory, and partially extended also to diluted models. In particular, we point out the structure of the variational principles for the free energy, that arise for these models.

##### ⊗ Christof Külske (University of Groningen)

###### What disorder can do to continuous interfaces

We consider statistical mechanics models of continuous
spins in a disordered environment. These models have a natural interpretation
as effective interface models. It is well known that without disorder there
are no interface Gibbs measures in infinite volume in dimension *d=2*,
while there are gradient Gibbs measures describing an infinite-volume distribution
for the increments of the field, as was shown by Funaki and Spohn.

In this talk we will present results describing the destabilizing effect of disorder on the interface, on the level of finite-volume interfaces, infinite-volume gradient states, and pinned interfaces.

(Joint work with A. van Enter and Enza Orlandi)

##### ⊗ Gady Kozma (Weizmann Institute)

###### The Alexander-Orbach conjecture

We show that random walk on the incipient infinite cluster of critical
percolation in high dimensions satisfies anomalous diffusion with spectral
dimension equal to *4/3*.

(Joint work with Asaf Nachmias)

##### ⊗ Ronald Meester (Vrije Universiteit Amsterdam)

###### Stabilizability in the abelian and in Zhang's sandpile model

We discuss the abelian sandpile and Zhang's sandpile models in infinite volume. One of the most basic questions is whether or not the avalanches lead to a stable configuration, and if the limiting configuration is unique. The answer depends on the initial configuration.

For the abelian sandpile, we show that the order in which topplings are performed does not matter
- if the initial configuration is stabilizable, then the limit is unique - and we show that in
dimension *1* there is a phase transition at initial density *1*. We also
prove that in dimension *1*, at the critical density, no stabilization occurs.

For Zhang's sandpile model, the situation is quite different. The order does matter: there
is no unique limit, and even the very stabilizability may depend on the order of the topplings.
In dimension *1*, below initial density *1/2* stabilizability
always occurs, above density *1* it never occurs, and we show by example that
in between, various things may happen.

(Various parts are joint work with different people: Frank Redig, Anne Fey den Boer and Haiyan Liu)

##### ⊗ Pierre Nolin (ENS & Université Paris-Sud)

###### Near-critical percolation and the geometry of diffusion fronts

We discuss a model of inhomogeneous medium known as *Gradient Percolation*,
which consists in a percolation process where the density of occupied sites depends on the
location in plane. This model was first introduced by the physicists Gouyet, Rosso and Sapoval
in 1985 to show that diffusion fronts are fractal. The macroscopic interface - separating occupied
sites and vacant sites - that appears remains localized in regions where the density of occupied
sites is close to the percolation threshold *p _{c}*, its behavior
can thus be described using properties of near-critical standard percolation. We then study a simple
two-dimensional model where a large number of particles that start at some site diffuse independently.
As the particles evolve, a concentration gradient appears and we observe a macroscopic interface: we
exhibit a regime where this (properly scaled) interface is fractal with dimension

*7/4*, as predicted by physicists.

##### ⊗ Vladas Sidoravicius (IMPA & CWI)

###### Diffusion limited aggregation and recurrence of Markov chains

##### ⊗ Murad Taqqu (Boston University)

###### Self-similarity and computer network traffic

In this lecture we will introduce self-similarity in the context of computer network traffic. It will show why self-similarity is important in this area.

Ethernet local area network traffic appears to be approximately
statistically self-similar. This discovery, made about eight years ago,
has had a profound impact on the field. I will try to explain what
statistical self-similarity means, how it is detected and indicate how
one can construct random processes with that property by aggregating a
large number of *on-off* renewal processes. If the number of
replications grows to infinity then, after rescaling, the limit turns
out to be the Gaussian self-similar process called fractional Brownian
motion. If, however, the rewards are heavy-tailed as well, then the
limit is a stable non-Gaussian process with infinite variance and
dependent increments. Since linear fractional stable motion is the
stable counterpart of the Gaussian fractional Brownian motion, a
natural conjecture is that the limit process is linear fractional
stable motion. This conjecture, it turns out, is false. The limit is
a new type of infinite variance self-similar process.

(Joint work with Walter Willinger, Vladas Pipiras and others)

##### ⊗ Remco van der Hofstad (TU Eindhoven)

###### Universality of distances in random graphs

The topological properties of complex networks have
received enormous attention. Measurements have shown fascinating
common features, such as power law degree sequences and the *small
world phenomenon*. The small world phenomenon states that
typical distances in large complex networks are small, while
a complex network has a power law degree sequence when the number
of vertices with degree proportional to *k* decays as an
inverse power of *k*.

We discuss several models for complex networks where the number
of vertices of the graph with degree *k* decays as an inverse
power of *k*. Such models can either be static, i.e., of
fixed size, or dynamic, i.e., growing with time. A much investigated
static model is the configuration model, while dynamic
models receiving substantial attention are preferential
attachment models, where the growth favors vertices which
already have high degree. We investigate distances in
such graphs. Perhaps surprisingly, the results suggest that
graph distances grow in a similar way for the models under
consideration when they have similar degrees. This hints at
a strong form of universality.

(Joint work with Gerard Hooghiemstra, Piet Van Mieghem, Dmitri Znamenski and Henri van den Esker)

#### Short presentations

##### ⊗ Luca Avena (Universiteit Leiden)

###### Annealed Large Deviation for a Random Walk in a Dynamic Random Environment

##### ⊗ Marco Aymone (IMPA)

###### Scars on quantum graphs

Quantum graphs (metric graphs equipped with the laplacian that acts on each edge of the graph) are a good model for understanding the relation between quantum and classical mechanics. More precisely, we are interested in the behavior of a simple quantum system in the limit of very large energies, when one expects to recover the classical behavior. From an applied point of view, it is also possible to say that quantum graphs represent the behavior of very small currents on nanocircuits, and its understanding is important in order to improve the design of this kind of device.

Here we would like to discuss an approach to the problem of existence of scars for certain quantum graphs (i.e. subgraphs that concentrate the probability associated to the square-moduli of the eigenfunctions over a certain subsequence of eigenvalues) that gives a new proof of a result obtained by Keating et al. in the context of the so called star graph.

Furthermore, we would like to show necessary conditions for a certain subgraph of a generic graph to be a scar, as well as some consequences of the existence of certain scars on the spectrum of the graph.

##### ⊗ Andras Balint (Vrije Universiteit)

###### Criticality in the Ising model

We show that the high temperature Ising model on the triangular
lattice is a critical percolation model as follows.
The Ising model at inverse temperature *β* and zero
external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model
with *q=2* and density of open edges
*p=1-e ^{-β}*
by assigning spin

*+1*or

*-1*to each vertex in such a way that

- all the vertices in the same FK cluster get the same spin
*+1*and*-1*have equal probability.

We generalize the above procedure by assigning spin *+1* with
probability *r* and *-1* with probability *1-r*,
with *r∈[0,1]*, while keeping condition (1).

For fixed *β*, this generates a dependent (spin)
percolation model with parameter *r*. We show that, on the triangular
lattice and for *β<β _{c}*,
this model has a sharp percolation phase transition at

*r=1/2*, corresponding to the Ising model.

(Joint work with Federico Camia and Ronald Meester)

##### ⊗ Luca De Sanctis (n. a.)

###### Dilute mean field ferromagnet

We introduce a mean field dilute ferromagnet and study rigorously its thermodynamic behavior. We gain control of the high temperature region and compute the entropy and the free energy density at temperature zero.

##### ⊗ Emilio De Santis (Università di Roma La Sapienza)

###### Continuity of oriented percolation probability for interacting models in two dimension

Under some general conditions, we prove the continuity of oriented percolation probability for interacting fields in dimension two. We show that none of the hypotheses can be dropped.

##### ⊗ Alessandra Faggionato (Università di Roma La Sapienza)

###### Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes

We consider *PDMPs* with a finite set of discrete states. In the regime
of fast jumps between discrete states, we state a law of large number and a large deviation
principle. In the regime of fast and slow jumps, we analyze a coarse-grained process associated
to the original one and show its convergence to a new *PDMP* with effective force fields
and jump rates.

We discuss some applications related to the mechanochemical cycle of macromolecules, including strained-dependent power-stroke molecular motors.

(Joint work with D. Gabrielli and M. Ribezzi)

##### ⊗ Michele Gianfelice (Università della Calabria)

###### Quantum methods for interacting particle systems

##### ⊗ Onur Gun (Courant Institute)

###### Extremal processes as a universal aging scheme for trap models

We give a general proof of (maximal) aging for trap models where
the trapping landscape is asymptotically fat-tailed. The proof is based on abstract
conditions on the potential theory of the underlying graph and on the randomness of
the trapping landscape. We apply this proof for trap dynamics of the *Random Energy
Model* on a broad range of time scales.

##### ⊗ Matthijs Joosten (Vrije Universiteit Amsterdam)

###### Trivial, critical and near-critical scaling limits in *2D*-percolation

Consider the two-dimensional hexagonal lattice. Color each hexagon *white*
with probability *p* and *black* with probability *1-p*, independently
of each other. We are interested in the edges between hexagons with different colors. These edges
form a collection of loops and we will study its scaling limit when the mesh of the lattice is
sent to zero. Using standard results it can be proven that there only three possible types for
the scaling limit, depending on the value of *p*.

##### ⊗ Noemi Kurt (Universität Zurich)

###### Entropic repulsion for a Gaussian interface model

We consider a Gaussian interface model on the
*d*-dimensional integer lattice, with covariances given by the
Green's function of the discrete Bilaplacian. We discuss the
effect of a forbidden region on the interface, and present
detailed results in the critical and supercritical dimensions
for this model.

##### ⊗ Mauro Mariani (Université de Paris Dauphine)

###### Large deviations for Itô diffusions

We introduce a technique to investigate large deviations principles for a wide class of (finite and infinite dimensional) Itô diffusions, reducing the large deviations principle to the analysis of a variational deterministic problem. Some applications are discussed.

##### ⊗ Jason Miller (Stanford University)

###### Quasi-stationary random overlap structures and the continuous RPC

A random overlap structure (*ROSt*) is a measure on pairs
*(X,Q)* where *X* is a locally finite sequence in
* R* with a maximum and

*Q*a positive semidefinite matrix of overlaps intrinsic to the particles

*X*.

Such a measure is said to be quasi-stationary provided that the joint law of the gaps
of *X* and overlaps *Q* is stable under a stochastic evolution
driven by a Gaussian sequence with covariance *Q*.

Aizenman et al. have shown that quasi-stationary *ROSts* serve as an important
computational tool in the study of the Sherrington-Kirkpatrick (*SK*) spin-glass
model from the perspective of cavity dynamics and the related *ROSt* variational
principle for its free energy. In this framework, the Parisi solution is reflected in
the ansatz that the overlap matrix exhibits a certain hierarchical structure. Aizenman
et al. posed the question of whether the ansatz could be explained by showing that the
only *ROSts* that are quasi-stationary in a robust sense are given by a special
class of hierarchical *ROSts* known as both the *Ruelle Probability Cascades*
as well the *GREM*. Arguin and Aizenman recently gave an affirmative answer in
the special case that the set of entries *S _{Q}* of

*Q*is finite.

We will show show that this result holds more generally when
*|S _{Q}|=∞* provided that

*Q*satisfies the technical condition that

*S*has no limit points from below.

_{Q}##### ⊗ Ida Germana Minelli (Università di Padova)

###### Monotonicity and complete monotonicity for continuous-time Markov chains

##### ⊗ Leonardo Rolla (IME-USP & IMPA)

###### Phase transition for activated random walk models

The model of activated random walks evolves as follows.
On * Z^{d}*, at time
zero there is an

*iid*number of active particles whose expectation is

*μ*. Each particle performs a continuous-time random walk with rate one. When a particle is found alone at some site, it will change its state from active to inactive when an exponential clock of rate

*λ*rings. Once a particle is inactive, it no longer jumps. When some other particle jumps into the same site, the particle becomes active again.

A natural conjecture is that for low *μ* the
system locally fixates and for large *μ* there is no
fixation. We identify the important variables that charactherize fixation,
are monotone on the parameters and introduce an explicit graphical representation
that couples these variables in the appropriate way. Such representation
has a very useful commutativity property that favors algorithmic approaches.
With this method we prove that there is a (unique) phase transition for the
one-dimensional finite-range random walk.

(Joint work with V. Sidoravicius)

##### ⊗ Artem Sapozhnikov (CWI)

###### Invasion percolation in 2D (Part II)

My talk is a continuation of the talk by Balint Vagvolgyi.
I will present further results on the distribution of the invasion ponds. In
particular, I will show that the probability that the diameter of the
*k ^{th}* pond is bigger than

*n*is comparable with the probability that there is a

*p*-open path with

_{c}*(k-1)*defects from the origin to the boundary of

*B(n)*.

(Joint work with Michael Damron and Balint Vagvolgyi)

##### ⊗ Mykhaylo Shkolnikov (Stanford University)

###### Competing particle systems evolving by i.i.d. increments

We consider competing particle systems in
* R^{d}*,
i.e. random locally finite upper bounded configurations of points in

*evolving in discrete time steps. In each step i.i.d. increments are added to the particles independently of the initial configuration.*

**R**^{d}Ruzmaikina and Aizenman characterized quasi-stationary measures of
such an evolution, i.e. point processes for which the joint distribution
of the gaps between the particles is invariant under the evolution,
in case *d=1* and restricting to increments having a density.

We prove a multidimensional version of the *Ruzmaikina-Aizenman Theorem*
which extends beyond the density case requiring only a growth condition
on the increments and assuming that under the initial configuration no
two particles are located at the same point. The results may be of interest
in the context of non-Gaussian spin glass models as the Ruzmaikina-Aizenman
Theorem is for the Sherrington-Kirkpatrick model of spin glasses.

##### ⊗ François Simenhaus (Paris VII)

###### Random walk delayed on percolation clusters

We study a continuous time random walk on the *d*-dimensional lattice, subject
to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. More precisely,
given such a percolation we define a continuous time markov chain with a drifted random walk as skeleton
and jump rates given by *e ^{βCx}* (where

*C*denotes the size of the cluster of

_{x}*x*). This model can be seen as a

*Bouchaud Trap Model*modified in the sense that the jump rates are not i.i.d.

We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.

##### ⊗ Balint Vagvolgyi (Vrije Universiteit)

###### Invasion percolation in 2D (part I)

Invasion percolation is a stochastic growth model, closely related to critical Bernoulli percolation.
One possible interpretation is the following: consider an infinite piece of land that is divided into square parcels
separated by dikes on the four sides. The dikes have independent, *uniform(0,1)* distributed
heights. One of the parcels contains an infinite source of water. As the water keeps flowing from the source it will
eventually reach the level of the smallest dike and overflow into the neighbouring parcel and so on. The union of the
parcels reached by the flood as the time goes to infinity is called the invaded region.

The main interest in invasion percolation arises from its self-organized critical nature. In this talk, I am going introduce the invasion model and define the ponds of the invaded region. Recent results comparing the size of these ponds to that of critical clusters in Bernoulli percolation will be presented in the continuation of this talk by Artem Sapozhnikov.

(Joint work with Michael Damron and Artem Sapozhnikov)