## Workshop 2

Spin Glasses, Random Graphs and Percolation

February 16

^{th}to 20^{th}Participants who intend to present a poster can submit their request to the Poster Session Chair (Prof. Claudio Giberti, *claudio.giberti@unimore.it)*

Invited speakers:

*D. Alberici*“Monomer-Dimer model on a class of random graphs”, [abstract]
I will introduce the Monomer-Dimer model in Statistical
Mechanics, and show its solution on the Erdos-Rényi random graph
(based on a joint work with P. Contucci). This result relies on the
fact that such a graph locally looks like a Galton-Watson tree. Beyond
this theorem of Probability Theory, the main ingredients in the proof
of our result are the analiticity of the Monomer-Dimer model together
with some alternating correlation inequalities.

*L.P. Arguin*“Maxima of log-correlated Gaussian fields and of the Riemann Zeta function on the critical line”, [abstract]
A recent conjecture of Fyodorov, Hiary & Keating states that the
maxima of the Riemann Zeta function on a bounded interval of the
critical line behave similarly to the maxima of a specific class of
Gaussian fields, the so-called log-correlated Gaussian fields. These
include important examples such as branching Brownian motion and the
2D Gaussian free field. In this talk, we will highlight the
connections between the number theory problem and the probabilistic
models. We will outline the proof of the conjecture in the case of a
randomized model of the Zeta function. We will discuss possible
approaches to the problem for the function itself.
This is joint work with D. Belius (NYU) and A. Harper (Cambridge).

*G. Ben Arous*“Scaling limit for the ant in a labyrinth”, [abstract]
In an ongoing work with Alexander Fribergh and Manuel Cabezas, we
give a natural scaling limit for the Random Walk on a simple critical
percolation structure, i.e. large critical branching random walks, in high
enough dimension. If time permits, I will also describe the scaling limit
for the walk projected on the backbone for the incipient infinite cluster
and shows how it relates to the specific SSBM, or spatially subordinated
brownian motion, which was obtained as a scaling limit in the case of the
random walk on the incipient critical tree, in a recent work with
M.Cabezas, and formerly introduced as limits of 1d trap models in a joint
work with M.Cabezas, R. Royfman and J.Cerny.

*A. Bovier*“Extremal Processes of Gaussian Processes Indexed by Trees”, [abstract]
Gaussian processes indexed by trees form an interesting class of
correlated random fields where the structure of extremal processes can be
studied. One popular example is Branching Brownian motion, which has received
a lot of attention over the last decades, non the least because of its
connection to the KPP equation.
In this talk I review the construction of the extremal process of standard
and variable speed BBM (with Arguin, Kistler, and Hartung).

*W.K. Chen*“On the uniqueness and properties of the Parisi measure”, [abstract]
Spin glasses are disordered spin systems originated from the
desire of understanding the strange magnetic behaviors of certain alloys in
physics. As mathematical objects, they are often cited as examples of
complex systems and have provided several fascinating structures and
conjectures. This talk will be focused on one of the most famous mean-field
spin glasses, the Sherrington-Kirkpatrick model. We will present results on
the conjectured properties of the Parisi measure including its uniqueness
and quantitative behaviors. This is based on joint works with A. Auffinger.

*F. Den Hollander*“Annealed Scaling for a Charged Polymer”, [abstract]
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 in higher dimensions remains largely open. We report on some preliminary results.

Joint work with Q. Berger, F. Caravenna, N. Petrelis and J. Poisat.

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 in higher dimensions remains largely open. We report on some preliminary results.

Joint work with Q. Berger, F. Caravenna, N. Petrelis and J. Poisat.

*H. Duminil-Copin*“A new proof of exponential decay of correlations in subcritical percolation and Ising models”, [abstract]
We provide a new proof of exponential decay of correlations for
subcritical Bernoulli
percolation on Z^d. The proof is based on an alternative definition of the
critical point.
The proof extends to the Ising model and to infinite-range models on
infinite locally-finite transitive graphs. It also provides a mean-field
lower bound for the
explosion of the infinite-cluster density in the supercritical regime.

Joint work with Vincent Tassion.

Joint work with Vincent Tassion.

*C. Giberti*“Limit theorems for Ising models on random graphs”, [abstract]
In this talk I will present some limit theorems for the total
spin of ferromagnetic Ising models
on random graphs. In such systems two different source of randomness are
present: the Ising spins that are random variables distributed according
to the Boltzmann-Gibbs measure and the spatial disorder given by the random
graph. This is the source of distinction between the "random quenched
measure", the "average quenched measure" and the "annealed measure". In
the uniqueness regime,
the Central Limit Theorem (CLT) can be proven for the general tree-like
graphs in the random quenched setting. The average quenched CLT is much
more challenging, thus we restrict ourselves to two "configuration models":
the model with all vertices having degree 2 (CM(2)) and the case where
vertices have either degree 1 or degree 2 (CM(1,2)). In the annealed
setting the CLT can be proven for the Ising model on the CM(2), CM(1,2)
graphs and also on the Generalized Random Graph. This last model has a
finite critical temperature. We will discuss the role of the fluctuation of
the vertex degree in the limit normal behaviors.

*F. Guerra*“Legendre structures in statistical mechanics”, [abstract]
We review some general properties of statistical mechanics systems
leading to Legendre structures. The primary origin of these structure
is the second principle of thermodynamics. However, they acquire
a very ubiquitous presence in the case of the various models.
In particular we discuss the case of multi-species ferromagnetic models,
spin glasses and neural networks. The connection with the order parameters
is pointed out. In particular we give the interpretation of the Parisi
functional order parameter in this frame, and point out some
generalizations.

*R. van der Hofstad*“Competition on scale-free random graphs”, [abstract]
Empirical findings have shown that many real-world networks share
fascinating features. Indeed, many real-world networks are small worlds, in
the sense that typical distances are much smaller than the size of the
network. Further, many real-world networks are scale-free in the sense that
there is a high variability in the number of connections of the elements of
the networks, making these networks highly inhomogeneous. Such networks are
typically modeled using random graphs with power-law degree sequences.

In this lecture, we will investigate the behavior of competition processes on scale-free random graphs with finite-mean, but infinite-variance degrees. Take two vertices uniformly at random, or at either side of an edge chosen uniformly at random, and place an individual of two distinct species at these two vertices. Equip the edges with traversal times, which could be different for the two species. Then let each of the two species invade the graph, such that any other vertex can only be occupied by the species that gets there first. Let the speed of the species be the inverse of the expected traversal times of an edge by that species.

We distinguish two cases. When the traversal times are exponential, we see that one (not necessarily the faster) species will occupy almost all vertices, while the losing species only occupied a bounded number of vertices, i.e., the winner takes it all, the loser's standing small. In particular, no asymptotic coexistence can occur. On the other hand, for deterministic traversal times, the fastest species always gets the majority of the vertices, while the other occupies a subpolynomial number. When the speeds are the same, asymptotic coexistence (in the sense that both species occupy a positive proportion of the vertices) occurs with positive probability.

This lecture is based on joint work with Mia Deijfen, Julia Komjathy and Enrico Baroni, and builds on earlier work with Gerard Hooghiemstra, Shankar Bhamidi and Dmitri Znamenski.

In this lecture, we will investigate the behavior of competition processes on scale-free random graphs with finite-mean, but infinite-variance degrees. Take two vertices uniformly at random, or at either side of an edge chosen uniformly at random, and place an individual of two distinct species at these two vertices. Equip the edges with traversal times, which could be different for the two species. Then let each of the two species invade the graph, such that any other vertex can only be occupied by the species that gets there first. Let the speed of the species be the inverse of the expected traversal times of an edge by that species.

We distinguish two cases. When the traversal times are exponential, we see that one (not necessarily the faster) species will occupy almost all vertices, while the losing species only occupied a bounded number of vertices, i.e., the winner takes it all, the loser's standing small. In particular, no asymptotic coexistence can occur. On the other hand, for deterministic traversal times, the fastest species always gets the majority of the vertices, while the other occupies a subpolynomial number. When the speeds are the same, asymptotic coexistence (in the sense that both species occupy a positive proportion of the vertices) occurs with positive probability.

This lecture is based on joint work with Mia Deijfen, Julia Komjathy and Enrico Baroni, and builds on earlier work with Gerard Hooghiemstra, Shankar Bhamidi and Dmitri Znamenski.

*M. Holmes*“WARM graphs”, [abstract]
We consider a class of reinforcement processes in which there are
n colours of balls and at each iteration of the process one selects some
subset of the colours to compete against each other, reinforcing the
winner. When defined on a graph this is a toy model for reinforcement in
the brain, and it is natural to consider the random subgraph of colours
(edges) selected a positive proportion of the time. [Joint with Remco van
der Hofstad, Alexey Kuznetsov, and Wioletta Ruszel]

*D. Ioffe*“A quantitative Burton-Keane estimate under strong FKG condition”, [abstract]
We consider translation-invariant percolation models
on $Z^d$ satisfying the finite energy and the FKG properties.
We provide explicit upper bounds on the probability of having two distinct
clusters going from the endpoints of an edge to distance $n$ (this
corresponds to a finite size version of the celebrated Burton-Keane
argument proving uniqueness of the infinite-cluster). The proof is based
on the generalization of a reverse Poincar\'e inequality for Bernoulli
percolation which was derived by Chatterjee and Sen.
As a consequence, we show how RSW-type estimates recently obtained by
Duminil-Copin, Sidoravicius and Tassion imply upper bounds on the
probability of the so-called four-arm event for planar random-cluster
models with cluster-weight $q\in [1,4]$.

Joint work with Hugo Duminil-Copin and Yvan Velenik

Joint work with Hugo Duminil-Copin and Yvan Velenik

*N. Kistler,*“A multiscale refinement of the second moment method”, [abstract]
I will discuss a version of the second moment method
which is particularly efficient to analyze the extremes of random
fields where multiple scales can be identified. The method emerged
from work on the extremes of branching Brownian motion, joint
with Louis-Pierre Arguin (CUNY) and Anton Bovier (Bonn), and from
a recent work with David Belius (NYU) on the cover time by planar
Brownian motion. Time permitting, I will also discuss a procedure of
local projections which allows, in a number of models, to generate
scales from first principles.

*E. Mingione*“A Monomer-Dimer model with random weights on the complete graph”, [abstract]
In this talk we considered a monomer-dimer model with
independent random
weights on the complete graph. Under very general conditions on the randomness
one can prove that the quenched pressure density is self-averaging.
In the case of i.i.d. monomer random weights we prove that the
thermodynamical limit of the quenched pressure is given by the
solution of a one-dimensional variational principle. We show that such
solution is a smooth function of the dimeric weight.
A sketch of the proof, based on a gaussian representation of the
partition function, and some open questions will be discussed.

*D. Panchenko*“Chaos in temperature in generic 2p-spin models”, [abstract]
I will discuss a proof of chaos in temperature for even
p-spin models which include sufficiently many p-spin interaction
terms. The approach is based on a new invariance property for coupled
asymptotic Gibbs measures, similar in spirit to the invariance
property that appeared in the proof of ultrametricity, used in
combination with Talagrand's analogue of Guerra's replica symmetry
breaking bound for coupled systems.

*A. Sakai*“Critical correlation in high dimensions for long-range models with power-law couplings”, [abstract]
It is known that percolation with finite-range symmetric bond occupation
probability p(x,y) exhibits mean-field behavior above the upper-critical
dimension 6. The situation is the same even if we modify p(x,y) to decay
as |x-y|^{-d-a} with a>2. If 0<a<2, then the second moment of p diverges
and the situation changes: the upper-critical dimension becomes 3a, and
the critical two-point function decays like the Riesz potential, not like
the Newtonian potential as in the case of a>2. The marginal case of a=2
has not been studied before, even the asymptotic expression of the random
walk's Green function is not known. I will explain the ongoing work (with
Lung-Chi Chen) on this marginal case, and outline the proof of the
asymptotic expression of the critical two-point function, which turns out
to have a log correction to the Newtonian potential, for dimensions above
or equal to 6.

*S. Starr*“Spin glass techniques for non-Hermitian random matrices”, [abstract]
Spin glass techniques include finding consequences of
self-averaging properties such as stochastic stability, and identifying the
distributional limit of overlaps between two configurations drawn from the
equilibrium measure. For non-Hermitian random matrices, such as the complex
Ginibre ensemble, there are analogous steps. A consequence of
self-averaging is a simple limiting formula for all mixed matrix moments.
But the overlap between two random eigenvectors is more complicated. Its
study was initiated by Chalker and Mehlig, but it is not completed. I will
describe this and also some physical applications, such as by Fyodorov and
coworkers.

*D. Stein*“Rugged Landscapes and Timescale Distributions in Complex Systems”, [abstract]
Disordered systems are universally characterized by a high
degree of metastability, with
the consequence that much of their dynamical behavior can be
mathematically recast as a random walk on a "rugged landscape".
Many models of such walks treat the landscape as effectively
one-dimensional. However, a more accurate representation is that of
a landscape in very high dimension. Rigorous results, to be discussed in
this talk, demonstrate that the random walk in an appropriate
limit maps to invasion percolation, and that this in turn leads to some
novel, and often surprising and unexpected, behavior.

*R. Sun*“Polynomial chaos and scaling limits of disordered systems”, [abstract]
We formulate general conditions ensuring that a sequence of
multi-linear polynomials of independent random variables (called polynomial
chaos expansions) converges to a limiting random variable, given by a
Wiener chaos expansion over the d-dimensional white noise. This provides a
unified framework to study the continuum and weak disorder scaling limits
of statistical mechanics systems that are disorder relevant, including the
disordered pinning model with renewal exponent between 1/2 and 1, the
(long-range) directed polymer model in dimension 1+1, and the
two-dimensional random field Ising model. This gives a new perspective in
the study of disorder relevance, and leads to interesting new continuum
models. However, this approach breaks down for systems where the disorder
is marginally relevant, such as the disordered pinning model with renewal
exponent 1/2, and the directed polymer model in dimension 2+1.
Nevertheless, we show that a scaling limit still exists, and is furthermore
universal across the different models. Joint work with Francesco Caravenna
and Nikos Zygouras.

*F. Toninelli*“A class of (2+1)-dimensional growth process with explicit stationary measure”, [abstract]
We introduce a class of (2 + 1)-dimensional random growth processes,
that can be seen as asymmetric random dynamics of discrete interfaces.
Interface configurations correspond to height functions of dimer
coverings of the infinite hexagonal or square lattice. “Asymmetric”
means that the interface has an average non-zero drift. When the
asymmetry parameter p − 1/2 equals zero, the infinite-volume Gibbs
measures pi_\rho (with given slope \rho) are stationary and
reversible. When p\neq 1/2, \pi_\rho is not reversible any more but,
remarkably, it is still stationary. In such stationary states, one
finds that the height function at a given point x grows linearly with
time t with a non-zero speed, := <(hx(t) − hx(0))> = v t and
that the typical fluctuations of Q_x(t) are smaller than any power of
t.

For the specific case p = 1 and in the case of the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” studied by A. Borodin and P. L. Ferrari. For a suitably chosen initial condition (that is very different from the stationary state), they were able to determine the hydrodynamic limit and the interface fluctuations, exploiting the fact that some space-time correlations can be computed exactly.

For the specific case p = 1 and in the case of the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” studied by A. Borodin and P. L. Ferrari. For a suitably chosen initial condition (that is very different from the stationary state), they were able to determine the hydrodynamic limit and the interface fluctuations, exploiting the fact that some space-time correlations can be computed exactly.

*F. Zamponi*“Exact computation of the critical exponents of the jamming transition”, [abstract]
The jamming transition marks the emergence of rigidity in a system of
amorphous and athermal grains. It is characterized by a divergent
correlation length of the force-force correlation and non-trivial
critical exponents that are independent of spatial
dimension, suggesting that a mean field theory can correctly predict
their values. I will discuss a mean field approach to the problem
based on the exact solution of the hard sphere model in infinite
dimension. An unexpected analogy with the Sherrington-Kirkpatrick spin
glass model emerges in the solution: as in the SK model, the glassy
states turn out to be marginally stable, and are described by a Parisi
equation. Marginal stability has a deep impact on the critical
properties of the jamming transition and allows one to obtain analytic
predictions for the critical exponents. The predictions are consistent
with a recently developed scaling theory of the jamming transition,
and with numerical simulations. Finally, I will briefly discuss some
possible extensions of this approach to other open issues in the
theory of glasses.

*L. Zdeborova*“Percolation on sparse networks”, [abstract]
We reformulate percolation on networks as a message passing
process and demonstrate how the resulting equations can be used to
calculate, among other things, the size of the percolating cluster and the
average cluster size. The calculations are exact for sparse networks when
the number of short loops in the network is small, but even on networks
with many short loops we find them to be highly accurate when compared with
direct numerical simulations. By considering the fixed points of the
message passing process, we also show that the percolation threshold on a
network with few loops is given by the inverse of the leading eigenvalue of
the so-called non-backtracking matrix.