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Location: Aula Tricerri
Morning lecturer: Massimiliano Gubinelli (Bonn)
Title: Universality in slow growth phenomena and singular martingale problems
Abstract: In the last years there have been steady progress in understanding
the large scale properties of one dimensional growing interfaces. In the
regime where the growth is comparable to thermal fluctuations the interface
are described via the Kardar-Parisi-Zhang equation (KPZ). This is a stochastic
partial differential equation which contains a non-linearity whose precise
meaning is not apriori clear. There are various analytic approaches to make
sense of such an equation like regularity structures, paracontrolled
distributions, rough paths. In this talk we will describe a further approach
to the KPZ equation based on the probabilistic notion of martingale problem.
This approach can be used to prove the scaling limit of interface fluctuations
in a wide class of models. Due to the singular nature of the equation, the
martingale problem has to be formulated in a non-standard way and several new
ideas are needed to obtain a mathematically satisfactory theory. The aim of
the first part of talk will be to give a wide perspective on the phenomenon of
universality of the KPZ and related equations and in the issues involved in
their definition. In the second part we will discuss more in details the
well-posedness of the martingale problem, in particular the uniqueness
problem.
Afternoon lecturer: Julia Komjathy (Eindhoven)
Title: How to stop explosion by penalising transmission to hubs
Abstract:
In this talk we study the spread of information on infinite
inhomogeneous spatial random graphs.
We take a scale-free spatial random graph, where the degree of a vertex
follows a power law with exponent tau >1.
Examples of such graphs include: Scale free percolation, Geometric
Inhomogeneous Random Graphs, and Hyperbolic Random Graphs.
Then we equip each edge with a random and iid transmission delay L, and
study the ball-growth of the first-passage infected cluster around the
source vertex as a function of time. For a second, more realistic
spreading model, the iid random transmission delay L through an edge with
expected degrees W and Z is multiplied by a factor that is a polynomial of
W,Z, (the penalty factor).
We call the model outwards (inwards) explosive if it is possible to reach
infinitely many vertices within finite time (if infinitely many vertices
can reach a target vertex within finite time).
We will discuss the criterion for explosion in the original model (no
penalty factor) and in the penalised model. In particular, we will discuss
that asymmetric penalty functions can lead to `outwards' explosion but no
`inwards' explosion or the other way round.
Joint work with John Lapinskas and Johannes Lengler.
November 22nd, 2019
Location: Aula Tricerri
Morning lecturer: Sabine Jensen (Munich)
Title: Large deviations and metastability for the Widom-Rowlinson model
Abstract: The Widom-Rowlinson model is one of the few models in statistial
mechanics for which a phase transition is rigorously proven. It is also
popular in stochastic geometry and spatial statistics where it is called
area interaction model and belongs to the broader class of quermass
interaction models. After reviewing some relevant background about Gibbs
measures for continuum interacting particle systems, I will discuss large
deviations for the Widom-Rowlinson model in a joint high-density /
low-temperature limit. I will also discuss metastability for a spatial
birth and death process, a.k.a. continuum Glauber or grand-canonical
Monte-Carlo, for which the Gibbs measure is reversible. Based on joint
work with Frank den Hollander, Roman Kotecký and Elena Pulvirenti.
Afternoon lecturer: Luca Avena (Leiden)
Title: Explorations of networks through random spanning forests: theory
and applications
Abstract:
David Wilson in the 1990s described a simple and efficient
algorithm based on loop-erased random walks to sample uniform spanning
trees and, more generally, weighted rooted trees or forests spanning a
given graph.
The goal of this lecture is to describe the resulting probability measure
when Wilson's algorithm is used to sample rooted spanning forests.
This forest-measure has a rich, flexible and explicit mathematical
structure which makes it a powerful tool to design different algorithms to
explore a given network.
In the first part of the lecture, I will focus on fundamental aspects of
this measure and how it relates to other objects of interest in
statistical physics such as the well known Random-cluster model.
I will in particular describe the main properties of related observables
(e.g. set of roots, induced partition) which turn out to be determinantal
processes with simple kernels and then discuss some progress in
understanding related scaling limits.
The second part of the lecture will be devoted to applications. In
particular, depending on time, I plan to discuss four different algorithms
aiming at: (1) sampling well-distributed points in a graph,
(2) coarse-graning a given network, (3) processing signals on graphs (a
novel gaph wavelet transform), (4) estimating the spectrum of the graph
Laplacian.
The core of this lecture is based on different joint collaborations with
the following colleagues and students: Castell, Gaudillere, Melot,
Milanesi (Marselle), Quattropani (Rome), Driessen, Koperberg, Magrini
(Leiden) , Amblard, Barthelme, Tremblay (Grenoble).
September 27th, 2019
Location: Aula Tricerri
Morning lecturer: Pierre Picco (Marseille)
Title: One-dimentional Ising model with long range interactions. A review of
results.
Abstract:
In the first talk I will make an quick historical survey of the rigorous
results
on the one-dimensional Ising model with long-range interactions.
A first part will be dedicated to uniqueness of the Gibbs states
(Ruelle (1968); Dobrushin (1968); Bricmont, Lebowitz & Pfister (1986))
and the regularity of the free energy when the decay of the potential is fast
+enough (Dobrushin (1973) Cassandro & Olivieri (1981)
and its extensions in particular Capocaccia, Campanino & Olivieri (1983).
A second part will be dedicated to the existence of phase transition starting
from the Kac-Thompson conjecture (1968)
the Dyson results (1969), the Frohlich \& Spencer result (1982), the Imbrie
result (1982) the Aizenmann, Chayes, Chayes & Newman
result on the Thouless effect (1988),
Imbrie & Newman result on the Berezinsky,
Kosterlitz & Thouless transition (1988).
A third part will be dedicated to present results in the phase transition
regime
that started with Frohlich & Spencer (1981), Cassandro, Ferrari, Merola &
Presutti (2001) and its extensions in particular
by Cassandro, Merola, Picco & Rosikov (2014) on the definition of an interface
and its fluctuations,
and on a Minlos & Sinai theorem on the phase separation problem by Cassandro,
Merola & Picco (2017).
In the second talk I will review heuristic arguments that were invoked to
conjecture the existence of a phase transition at low temperature in particular
the Landau argument.
I will present toy models where the fluctuation of interfaces and localisation
of the droplet in the Minlos & Sinai theory will be explained. I will give an
algorithmic
definition of one-dimensional contours of Cassandro, Ferrari, Merola & Presutti.
Afternoon lecturer: Rui Pires da Silva Castro (Eindhoven)
Title:Testing for the presence of communities in inhomogeneous random graphs
Abstract: Many complex systems can be viewed as a network/graph consisting of
vertices (e.g., individuals) connected by edges (e.g., a friendship relation).
Often one believes there is some sort of community structure, where some
vertices are naturally grouped together (e.g., more densely connected between
themselves than to the rest of the network). Much of the community detection
literature is concentrated around methods that extract communities from a given
network. Our goal in this work is different, and we attempt to understand how
difficult is it to determine if a network has real communities. Furthermore, we
are primarily interested in the case of small or very small communities, for
which many existing results and methods are not applicable.
We cast this problem as a binary hypothesis test, where the null model
corresponds to a graph without community structure, and the alternative model
almost the same, but it also includes a planted community - that is, a small
subset of the vertices has higher connection probability than under the null.
The main question is to determine the minimal size and “strength” of the
planted community that will allow detection. The seminal work of Arias-Castro
and Verzelen tackled this problem when the null model is a homogeneous random
graph. In our work, however, we consider the case where the null model is
inhomogeneous, as this is somewhat closer to realistic scenarios. In
particular, we present a scan test and provide conditions under which it is
able to detect the presence of a small community. These results are valid for a
wide variety of parameter choices. Furthermore, we show that for some
parameters choices the scan test is optimal, and no other test can perform
better (e.g, detect smaller or weaker planted communities). Finally, we extend
this scan test to adapt to many parameters of the model when the null is a
rank-1 generalized random graph.
In the first part of the talk I will describe the above formulation and ensuing
results, with illustrative examples and briefly touching upon the analytical
methodology. In addition, I will discuss the related problem of characterizing
cliques in rank-1 random graphs, which provides some insights on the role of
inhomogeneity. The second part of the talk will go deeper into more technical
aspects and ensuing insights. This presentation is based on joint work with Kay
Bogerd and Remco van der Hofstad (https://arxiv.org/abs/1805.01688
and ongoing work).
March 22nd, 2019
Location: Aula Tricerri
Morning lecturer: Giovanni Gallavotti (Roma)
Title: Statistical ensembles, entropy and probability in statistical
mechanics, and extension to chaotic motions
(slides part one
and
slides part two
Abstract: a historical view on the theoretical developments generated by
Boltzmann's attempt to find a mechanical interpretation of the second
principle, from the action principle to the Boltzmann's equation to phase
transitions and their universality to the modern developments in the
non-equilibrium thermodynamics. In the second part the case of fluid
mechanics and an interpretation of viscosity and irreversibility will be
analyzed and related to an extension of the statistical ensembles to
non-equilibrium phenomena.
Afternoon lecturer: Silke Rolles (Technical University of Munich)
Title: Processes with reinforcement
(slides)
Abstract: In 1986, Persi Diaconis introduced edge-reinforced random
walk as a simple model for a tourist exploring an unknown city.
Already then, he raised the question of recurrence and transience
of this process on the d-dimensional integer lattice. Since
edge-reinforced random walk is more likely to traverse edges
that have been traversed often before and simple random walk is
recurrent in dimension 2, recurrence of edge-reinforced random
walk on the two-dimensional integer lattice may seem intuitively
clear. However, a proof of this result was only found in 2015
by Sabot and Zeng. For dimensions larger or equal to 3 a phase
transition between recurrence and transience was shown by
Disertori, Sabot and Tarres in 2011 and 2014.
In the talk I will give an overview of the subject and present
some basic techniques. In particular, the edge-reinforced random
walk is a mixture of reversible Markov chains with an explicitly
known mixing measure. In a special case, this can be illustrated
with an analogous result for the Polya urn.
Location: Aula Tricerri
Morning lecturer: Giovanni Jona Lasinio (Roma)
Title: Singular stochastic partial differential equations
(slides)
Abstract:
Singular stochastic partial differential equations (SSPDE) first
appeared in rather special contexts like the stochastic quantization of field
theories or in the problem of crystal growth, the well known KPZ equation. In
the last decade these equations have been intensely studied giving rise to an
important branch of mathematics possibly relevant for physics. This talk will
review some aspects and open problems in the subject.
Afternoon lecturer: Giambattista Giacomin (Paris)
Title: Infinite disorder renormalization fixed point: the big picture and one
specific result
(slides)
Abstract: the natural question of the effect of a random environment
(«disorder») on phase transitions and critical phenomena has attracted a lot of
attention. I will give an introduction to this domain of research via an
overview of some of the physical predictions and of the mathematical approaches
and challenges. I will in particular develop the notion of disorder relevance
and irrelevance. The focus will be on a very basic class of statistical
mechanics model - called pinning models - for which in the last years the
mathematical work matched the physical counterpart and, in some cases, went
beyond. Nevertheless, also for pinning models the results in the regime in
which disorder is relevant are rather weak and many of the physical predictions
do not appear to be solid or coherent. But the situation has evolved very
recently and a certain consensus has grown in favor of a very strong smoothing
effect of the disorder for this class of models when disorder is relevant. This
is part of a very intriguing and challenging general physical picture. The aim
of the second part of the seminar is to present a very specific pinning model
in which we have been able to pinpoint this strong smoothing effect (work in
collaboration with Quentin Berger and Hubert Lacoin, arXiv:1712.02261). I hope
I will be able to explain why we could tackle this case (and not other ones)
and to develop (or sketch) at least one of the main technical ideas that are at
the center of our approach.
Location: Aula Magna via San Gallo
Morning lecturer: Stefano Olla (Université Paris Dauphine)
Hyperbolic Hydrodynamic Limits
(slides)
Abstract:
I will present a review of old and new results (and open problems)
concerning scaling limits for conservation laws in the hyperbolic
space-time scale, for a system of anharmonic oscillators with external
boundary tension. The macroscopic equation is given by the compressible
Euler system, with corresponding boundary conditions. The problem is
particularly challenging when shockwaves are present.
Some results exists when the microscopic dynamics is perturbed by a
conservative stochastic viscosity. Works in Collaboration with Stefano
Marchesani (GSSI) and Lu Xu (CEREMADE).
Afternoon lecturer: Raú Rechtman (Universidad Nacional Autónoma
de México)
Title: Chaos and damage spreading in a probabilistic cellular automaton
Abstract:
Deterministic Boolean cellular automata (CA) are discrete
maps F:B^N -> B^N, B={0,1}, x(t+1)=F(x(t)) with x in B^N, N large and
t=
0,1,... . The vector x is the state of the cellular automaton with
components x[i], i=0,...,N-1 the state of cell I. Each cell is
connected to others, generally in a uniform and local way, and one can
define an adjacency matrix a[ij]=1 is cell j is connected to cell i and
zero otherwise. The global map F is determined by the parallel
application of a local function f, such that x[i](t+1) = f(v[i](t)),
where v[i] denotes the state of cells connected to cell i.
Deterministic CA are thus the discrete equivalent of dynamical systems,
and many concepts like trajectory (the sequence of configurations x
(t)), fixed points and limit cycles can be used. There are cellular
automata for which a small modification in an initial configuration
propagates to the whole system, a situation similar to chaos in
continuous systems, and indeed one can extend the concept of the
largest Lyapunov exponent to deterministic CA using Boolean
derivatives. One of the main inconvenient is that these systems do not
have continuous parameters to be tuned, in order to study bifurcations.
In probabilistic cellular automata, the function f (and thus F) is
defined in terms of transition probabilities so that deterministic CA
can be seen as the extreme cases of probabilistic ones, when the
transition probabilities are either zero or one. Probabilistic CA can
be seen also as Markov chains, and one can observe interesting phase
transitions after changing the transition probabilities that are
therefore continuous control parameters.
A realization of a specific trajectory is determined by the extraction
of one or more of random numbers for each cell. By extracting these
numbers at the beginning of the simulation, for all cells and all
times, one converts a probabilistic CA into a deterministic one,
running over a quenched random field. One can therefore use the
concepts of deterministic CA, like damage spreading and maximum
Lyapunov exponent also for probabilistic CA, with the advantage of
having the possibility of fine-tuning the control parameters.
In particular, we investigate a probabilistic cellular automaton which
can be considered an extension of a model in the universality class of
directed percolation models, but with two absorbing states. In the
first part of the talk all the concepts mentioned above are defined and
in the second part, the probabilistic cellular automaton is studied
numerically. We show that the phase transitions when the order
parameter is the average damage do not coincide with those found for
the Lyapunov exponent and the reason of this is the presence of
absorbing states.
Location: Aula Magna via San Gallo
Morning lecturer: Francis Comets (Université Paris-Diderot Paris 7)
Title: Cover time, cover process, random interlacements for random walk
on the torus
Abstract: to be announced
Afternoon lecturer: Remco van der Hofstad (TU/e, Eindhoven Technical University)
Title: Ising models on random graphs
Abstract:
The Ising model is one of the simplest statistical mechanics models that
displays a phase transition. While invented by Ising and Lenz
to model magnetism, for which the Ising model lives on regular lattices, it is
now widely used for other real-world applications as a model
for cooperative behavior and consensus between people. As such, it is natural
to consider the Ising model on complex networks. Since
complex networks are modelled using random graphs, this leads us to study the
Ising model on random graphs. In this talk, we discuss
some recent results on the stationary distribution of the Ising model on
locally tree-like random graphs. We start by giving an extensive introduction
to random graph models for complex networks, to set the stage of the graphs on
which our Ising models live. Real-world networks tend to be
highly inhomogeneous, a fact that is most prominently reflected in their degree
distributions having heavy tails as described by power laws.
Due to the randomness of the graphs on which the Ising model lives, there are
different settings for the Ising model on it.
The quenched setting describes the Ising model on the random graph as it is,
while the averaged quenched setting takes the expectation w.r.t.
the randomness of the graph. As such, it takes the expectation of the Boltzman
distribution, which is a ratio of an exponential involving
the Hamiltonian, and the partition function. In the annealed setting, the
expectation is taken on both sides of the ratio. These different settings
each describe different physical realities.
We discuss the thermodynamic limit of the Ising model, which can be used to
define the phase transition in the Ising model on locally tree-like
random graphs, by describing when spontaneous magnetization exists and when
not, extending work by Dembo and Montanari.
We give an explicit expression for the critical value and the critical
exponents for the magnetization close to it. These critical exponents
depend on the power-law exponent of the degree distribution in the random
graph. We also discuss central limit theorems for the total
spin in the uniqueness regime, as well as a non-classical limit theorem for the
total spin at the critical point in the special setting of the
annealed generalized random graph.
This talk is based on several joint works with Sander Dommers, Cristian
Giardina, Claudio Giberti and Maria Luisa Prioriello.
Location: Aula Magna via San Gallo
Morning lecturer: Frank den Hollander (Leiden University)
Title: Large deviations for the Wiener sausage
(slides)
Abstract: The Wiener sausage is the 1-environment of Brownian motion.
It is an important mathematical object because it is one of the simplest
non-Markovian functionals of Brownian motion. The Wiener sausage has been
studied intensively since the 1970's. It plays a key role in the study of
various stochastic phenomena, including heat conduction, trapping in random
media, spectral properties of random Schrödinger
operators, and Bose-Einstein condensation.
In these lectures we look at two specific quantities: the volume and the
capacity. After an introduction to the Wiener sausage, we show that both
the volume and the capacity satisfy a downward large deviation principle.
We identify the rate and the rate function, and analyse the properties of
the rate function. We also explain how the large deviation principles are
proved with the help of the skeleton approach.
Joint work with Michiel van den Berg (Bristol) and Erwin Bolthausen (Zurich).
Afternoon lecturer: Giovanni Peccati (University of Luxembourg)
Title: Stein's method and stochastic geometry
(slides)
Abstract:
The so-called 'Stein's method' for probabilistic approximations is a
collection of powerful analytical techniques, allowing one to explicitly assess
the distance between the distributions of two random objects, by using
caracterizing differential operators. Originally developed by Ch. Stein at the
end of the sixties for dealing with one-dimensional normal approximations under
weak dependence assumptions, Stein's method has rapidly become a crucial tool
in many areas of modern stochastic analysis, ranging from random matrix theory
and random graphs, to mathematical physics, geometry, combinatorics and
statistics. In the first part of my talk, I will provide a self-contained
introduction to Stein's method for normal approximations, by focussing on some
connection with generalised integration by parts formulae, both in a continuous
and discrete setting. In the second part of my talk, I will present some recent
applications of Stein's method in stochastic geometry, with specific emphasis
on the geometry of random fields, and on random geometric graphs.
Location: Aula Magna via San Gallo
Morning lecturer: Milton Jara (IMPA, Rio de Janeiro)
Title: Weak universality of the stationary KPZ equation
Abstract: A basic question about Markov chains is the asymptotic behavior of
integrals of some function of the chain along its trajectory. In the
literature, those integrals are sometimes called 'Birkhoff averages' or
'additive functionals'. In the first talk, I will introduce a general strategy
to estimate moment generating functions of these additive functionals in terms
of the relative entropy of the chain with respect to carefully constructed
reference measures. In the second talk, I will explain how to use this
strategy to prove that for a large class of weakly asymmetric stochastic
systems, the density of particles is well approximated by the stationary KPZ
equation. This proof does not require explicit knowledge of the stationary
measures of the stochastic systems, which was a major drawback of previous
results.
Afternoon lecturer: Nikos Zygouras (University of Warwick)
Title: Combinatorial structures in KPZ stochastic models
(slides)
Abstract: It was proposed by Kardar, Parisi and Zhang in the 1980s that a
large class of randomly growing interfaces exhibit universal fluctuations
described mathematically by a nonlinear stochastic partial differential
equation, which is now known as the Kardar-Parisi-Zhang or KPZ equation.
Examples of physical systems which fall in this class are percolation of
liquid in porous media, growth of bacteria colonies, currents in one
dimensional traffic or liquid systems, liquid crystals etc.
Surprisingly the fluctuations of such random interfaces are governed by
exponents and distributions that differ from the predictions given by the
classical central limit theorem and in dimension one are linked to laws
emerging from random matrix theory. The link between random growth and random
matrices (which still demands deeper investigation) is certain combinatorial
structures.
In these talks I review the current status of the field and describe some of
the combinatorial structures and the advances (both older and more recent)
that these have led to.
Location: Aula Magna via San Gallo
Morning lecturer: R. Fernandez (Mathematics Department, Utrecht)
Title: Signal description: process or Gibbs?
(slides)
Abstract: The distribution of signals such as spike trains is
naturally modeled through stochastic processes where the probability
of future states depend on the pattern of past spikes.
Mathematically, this corresponds to distributions *conditioned on the
past*. From a signal-theoretic point of view, however, one could
wonder whether a more efficient description could be obtained through
the simultaneous conditioning of past *and* future. Furthermore, such
a formalism could be appropriate when discussing string without a
particular "time" order, such as the distribution of DNA nucleotides,
or even issues related to anticipation and prediction in neuroscience.
On the mathematical level this double conditioning would correspond to
a Gibbsian description analogous to the one adopted in statistical
mechanics. In this talk I will introduce and contrast both approaches
-process and Gibbsian based- reviewing existing results on scope
and limitations of them.
Afternoon lecturer: Robert Morris (IMPA, Rio de Janeiro)
Title: Monotone cellular automata
(slides)
Abstract: Cellular automata are interacting particle systems whose
update rules are local and homogeneous. Since their introduction by von
Neumann almost 50 years ago, many particular such systems have been
investigated, but no general theory has been developed for their study,
and for many simple examples surprisingly little is known. Understanding
their (typical) global behaviour is an important and challenging problem
in statistical physics, probability theory and combinatorics.
In this talk I will outline some recent progress in understanding the
behaviour of a particular (large) family of monotone cellular automata
- those which can naturally be embedded in d-dimensional space -
with random initial conditions. For example, in the case where a site
updates (from inactive to active) if at least r of its neighbours are
already active, these models are known as bootstrap percolation, and
have been extensively studied for various specific underlying graphs.
Apart from their inherent mathematical interest, the study of these
processes is motivated by their close connection to models in
statistical physics, and I will discuss some applications to a family of
models of the liquid-glass transition known as kinetically constrained
spin models.
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