Content
Seminar in Probability Theory and Statistics  Past talks
Program FS 2021
Date/Time  Speaker  Title  Location 

4 January 2022 10:00  Nicolas Forien AixMarseille Université 
Some toy models of selforganized criticality based on percolation and on the Ising model (Abstract >)
The concept of selforganized criticality was coined in to describe some physical systems which present a criticallike behaviour, without the need to finely tune a parameter (like for example the temperature) to a particular value.
This idea has been put forward in various reallife settings (sand piles, forest fires, avalanches, neural networks, earthquakes...), with more or less controversial outcomes.
On the theoretical side, the rigorous construction and analysis of a mathematical model showing selforganized criticality is a difficult problem, and even models whose definition is very simple, like the sandpile model, are not well understood.
In this talk, I will present several toy models of selforganized criticality built upon percolation and the Ising model by introducing a feedback mechanism from the configuration onto the control parameter. The study of these models, which is a nice occasion to play around nearcritical phenomena, will lead us to compare with other related models and to discuss about the ingredients necessary to obtain a selfcritical behaviour. 
Zoom (link in EMail) 
Program HS 2021
Date/Time  Speaker  Title  Location 

15 September 2021 16:00  Benjamin McKenna NYU 
Complexity of Bipartite Spherical Spin Glasses (Abstract >)
Bipartite spin glasses  a variant of usual spin glasses, in which spins are grouped into two species  are a classical testing ground for new approaches in spin glasses. The annealed complexity of bipartite spherical models was initially considered by Auffinger and Chen, who gave upper and lower bounds. We give an exact variational formula for this complexity, both for pure spin glasses and for mixtures. We also find connections between this model and the usual spherical spin glasses, which were studied by Auffinger, Ben Arous, and Černý.

Zoom (link in EMail) 
22 September 2021 16:00  Pax Kivimae Northwestern University 
The GroundState Energy and Concentration of Complexity in Spherical Bipartite Models (Abstract >)
Bipartite spin glass models have been gaining popularity in the study of glassy systems with distinct interacting species. Recently, the annealed complexity of the pure spherical bipartite model was obtained by B. McKenna. In this talk, I will explain how to show that the lowlying complexity actually concentrates around this value, and how from this one can obtain a formula for the groundstate energy.

Zoom (link in EMail) 
20 October 2021 11:00  Antoine Maillard ETHZ 
Landscape Complexity for the Empirical Risk of inference models (Abstract >)
In this talk, I will introduce how we can use the KacRice method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents the first application of KacRice to obtain exact complexity asymptotics for nonGaussian random functions. We obtain a rigorous explicit variational formula for the average number of critical points. This result is then extended using the nonrigorous KacRice replicated method from theoretical physics. In this way we derive a second variational formula for the number of critical points for typical instances up to exponential accuracy. Finally, I will discuss some of the challenges ahead, in the numerical evaluation and extension of these formulas. This talk is based on the results of the paper https://arxiv.org/abs/1912.02143.

Kollegienhaus, Seminarraum 103 
22 December 2021 10:00  Jean Bernoulli Ravelomanana TU Dortmund 
Random KSAT (Abstract >)
TBA.

Zoom 
22 December 2021 13:00  Joon Lee Universität Frankfurt 
The Sparse Parity Matrix via Warning Propagation (Abstract >)  Zoom 
22 December 2021 16:00  Elizabeth CollinsWoodfin University of Michigan 
Spherical spin glass model with external field
(Abstract >)
This talk will focus on results from a recent paper by Baik, CollinsWoodfin, Le Doussal and Wu. In the paper, we analyze the free energy and the overlaps in the 2spin spherical Sherrington Kirkpatrick (SSK) spin glass model with an external field. Our goal is to understand the transition between this model and the one without an external field. My talk will provide a brief introduction to the SSK model, focusing on the distribution of spins (i.e. the geometry of the Gibbs measure) and how the spin distribution changes in the presence of an external field. I will then discuss the transitional case in which the strength of the external field goes to zero as the dimension of the spin variable grows. I will present results for overlaps with the external field, with the ground state and with a replica, focusing on what each of these overlaps tells us about the distribution of spins. Finally, I will provide a brief overview of our methods, including a contour integral representation of the partition function as well as random matrix techniques.

Zoom 
Program FS 2021
Date/Time  Speaker  Title  Location 

3 March 2021  Clayton Barnes Technion 
Effect of noise on the speed of the wavefront for
stochastic FKPP equations (Abstract >)
The FisherKPP equation is a well studied reactiondiffusion equation motivated by
biology as well as statistical physics. The solutions exhibit traveling "waves"
that propagate through space with a finite speed. Propagating waves also exist for
solutions to the stochastic FisherKPP equation (where the original FKPP equation
is perturbed by a white noise term). We determine the asymptotic behavior of the
wavespeed when the coefficient determining noisestrength approaches zero.

Zoom (link in EMail) 
24 March 2021  Fabrizio Leisen Univeristy of Nottingham 
Bayesian nonparametric conditional copula estimation of
twin data (Abstract >)
Several studies on heritability in twins aim at understanding the different
contribution of environmental and genetic factors to specific traits. Considering
the National Merit Twin Study, our purpose is to correctly analyse the influence of
the socioeconomic status on the relationship between twins’ cognitive abilities.
Our methodology is based on conditional copulas, which allow us to model the effect
of a covariate driving the strength of dependence between the main variables. We
propose a flexible Bayesian nonparametric approach for the estimation of
conditional copulas, which can model any conditional copula density. Our
methodology extends previous work by introducing dependence from a covariate in an
infinite mixture model. Our results suggest that environmental factors are more
influential in families with lower socioeconomic position.

Zoom (link in EMail) 
24 March 2021 16:00  Christian Brennecke Harvard University 
On the TAP equations for the SherringtonKirkpatrick
Model (Abstract >)
In this talk, I will review the ThoulessAndersonPalmer (TAP) equations for the
classical SherringtonKirkpatrick spin glass and present a dynamical derivation,
valid at sufficiently high temperature. In our derivation, the TAP equations follow
as a simple consequence of the decay of the two point correlation functions. The
methods can also be used to establish decay of higher order correlation functions.
We illustrate this by proving a suitable decay bound on the three point functions
which implies an analogue of the TAP equations for the two point functions. The
talk is based on joint work with A. Adhikari, P. von Soosten and H.T. Yau.

Zoom (link in EMail) 
14 April 2021 16:00  Shuangping Li Princeton 
Proof of the Contiguity Conjecture and Lognormal Limit
for the Symmetric Perceptron (Abstract >)
We consider the symmetric binary perceptron model, a simple model of neural
networks that has gathered significant attention in the statistical physics,
information theory and probability theory communities, with recent connections made
to the performance of learning algorithms. We establish that the partition function
of this model, normalized by its expected value, converges to a lognormal
distribution. As a consequence, this allows us to establish the contiguity
conjecture between the planted and unplanted models in the satisfiable regime and
other properties of the structure of the solution space. Our proof technique relies
on a dense counterpart of the small graph conditioning method, which was developed
for sparse models in the celebrated work of Robinson and Wormald.

Zoom (link in EMail) 
21 April 2021  Jack Bowden Univeristy of Exeter 
The Triangulation WIthin A STudy (TWIST) framework for
causal inference within Pharmacoepidemiological research (Abstract >)
Over the last 20 years the field of Epidemiology has embraced the exploitation of
random genetic inheritance to help uncover causal mechanisms of disease using the
technique of Mendelian randomization (MR). Genetic variants can also play an
important role in helping to explain treatment effect heterogeneity, through the
science of pharmacogenetics. A canonical example is Clopidogrel: the primary drug
for stroke prevention in the UK and many other countries. It requires CYP2C19
enzyme activation in order to be properly metabolised and thus work to its fullest
extent. We review the methodological underpinnings of the general pharmacogenetic
approach, which utilises only individuals who are treated and relies on fairly
strong baseline assumptions to estimate what we refer to as the `genetically
mediated treatment effect' (GMTE). When these assumptions are seriously violated,
we show that a robust estimate of the GMTE that incorporates information on the
population of untreated individuals can instead be used. In cases of partial
violation, we clarify when Mendelian randomization and a modified confounder
adjustment method can also yield consistent estimates for the GMTE. A full decision
framework is then described to decide when a particular estimation strategy is most
appropriate and how estimates can be combined to improve efficiency. We illustrate
these approaches by reanalysing UK BiobankCPRD linked data relating to CYP2C19
genetic variants, Clopidogrel use and stroke risk, and data relating to ApoE
genetic variants, statin use and Coronary Artery Disease. We then discuss how the
framework can be extended for use in other epidemiological contexts.

Zoom (link in EMail) 
28 April 2021 16:00  Francesco Concetti Univeristy of Arizona 
The full replica symmetry breaking in the Ising spin
glass on random regular graph (Abstract >)
This work focuses on the extension of the Parisi full replica symmetry breaking
solution to the Ising spin glass on a random regular graph. We propose a new
martingale approach that overcomes the limits of the ParisiMézard cavity method,
providing a welldefined formulation of the full replica symmetry breaking problem
in random regular graphs. We obtain a variational free energy functional, defined
by the sum of two variational functionals (auxiliary variational functionals), that
are an extension of the Parisi functional of the SherringtonKirkpatrick model. We
study the properties of the two variational functionals in detail, providing a
representation through the solution of a proper backward stochastic differential
equation, that generalize the Parisi partial differential equation. Finally, we
define the order parameters of the system and get a set of selfconsistency
equations for the order parameters and the free energy.

Zoom (link in EMail) 
5 May 2021  Guillaume ConchonKerjan Université de Paris 
Levelset percolation of the GFF on regular graphs:
emergence of a Gaussian giant (Abstract >)
The Gaussian Free Field, usually studied on transient graphs, can also be defined
on finite graphs (its covariance is given by a compensated Green function). This
talk is about the GFF on typical regular graphs whose number of vertices goes to
infinity. We will focus on the percolation of the GFF above a fixed level \(h\). If \(h\)
is below a certain critical threshold \(h_*\), we establish the emergence w.h.p. of a
unique giant component containing a positive proportion of the vertices, surrounded
by islets of logarithmic size, while only the latter survive if \(h\ge h*\). This big
continent shares many similarities with the giant component of the famous
ErdősRényi random graph.

Zoom (link in EMail) 
Friday 7 May 2021 9:30  Vladimir Fomichov Aarhus University 
Pfaffian structure and concentration of measure in coalescing particle systems (Abstract >)
In our talk, we will consider random point processes formed by the clusters in one
dimensional coalescing particle systems and connected to the real Ginibre ensemble.
Basing on recent work of R. Tribe and O. Zaboronski, we will first discuss their
Pfaffian structure in the Brownian case, and then, by proposing a new algebraic proof,
prove their conjecture that the obtained results remain valid in the general case.
Furthermore, we will speak about the concentration of measure in diffeomorphic approximations of Brownian coalescing particle systems. In particular, the highlevel exceedances of the stationary processes formed by the corresponding densities will be studied. More precisely, using the Rice formula, we will compute the levelcrossing intensity of these processes and establish its asymptotic behaviour as the height of the level tends to infinity. 
Zoom (link in EMail) 
Friday 7 May 2021 14:00  Vanessa Jacquier University of Florence 
Metastability for the Ising model on the hexagonal lattice (Abstract >)
We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the relaxation time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.

Zoom (link in EMail) 
12 May 2021  Björn Bornkamp Novartis 
Sensitivity analyses for principal stratum estimands (Abstract >)
Questions on the treatment effect in subpopulations defined by postrandomization events, are not uncommon
in drug development. In the causal inference literature these are typically referred to as principal stratum estimands.
As these populations are defined based on events that may be influenced by the received treatment, randomization
alone can no longer be relied upon for assessment of the treatment effect. Additional assumptions are required to identify
the estimand. As an example consider a multiple sclerosis trial, where interest may be in the treatment effect on
confirmed disability progression in the population that does not relapse during the trial. When the treatment prevents
relapses, it is difficult to perform this assessment, because the nonrelapser populations on treatment and control group
may not be comparable. As assumptions play an important role for identification of principal stratum estimands, sensitivity
analyses should play a crucial role too. I will review different approaches in this area and illustrate them in the context
of a simulated example data set and a recently proposed Bayesian analysis strategy.

Zoom (link in EMail) 
26 May 2021  Chiara Cammarota King's College 
Rough landscapes and glass dynamics: from inference to machine learning (Abstract >)
The realm of statistical mechanics has been enlarged to describe systems,
such as glass forming materials, where structural disorder plays the predominant role.
Interestingly the spectrum of applications of this new physics goes much beyond the
scope of condensed matter and extends to the currently booming field of data science.
In this talk I will focus on the challenge of signalreconstruction from noisy
collections of data, omnipresent in machine learning applications and in classical
inference problems. By leveraging tools and ideas from glass physics, I will give some
examples on how we can describe, predict, and enhance the performances of algorithms
introduced to tackle these reconstruction problems

Zoom (link in EMail) 
2 Jun 2021 10:3011:00 
Jean Barbier, Manuel Saenz ICTP Trieste 
TBA (Abstract >)
TBA

Zoom (link in EMail) 
9 Jun 2021 16:00  Zhehua Li Northwestern Univeristy 
Sharp asymptotics and trivialization phase of the (p,k) spiked
tensor model (Abstract >)
In this talk we focus on a class of disorder systems characterized by the spherical pure pspin Hamiltonian in the presence of a nonlinear signal given by a deterministic polynomial of degree k. (We will call them the (p,k) spiked tensor model). It is known recently that these systems exhibit phase transitions in terms of the topology of their energy landscape. We characterize the mean number of deep minima near the bottom of the landscape through the KacRice formula. When the strength of the signal is beyond a conjectured trivialization threshold, the mean number of deep minima is asymptotically finite (but is surprisingly not equal to 1) and we derive an explicit formula for the limiting ground states and the limiting ground state energy by combining techniques in spin glass theory and random matrices. This talk is based on joint work with Antonio Auffinger and Gerard Ben Arous.

Zoom (link in EMail) 
23 Jun 2021  Paul Bourgade Courant Institute 
Complexity of the elastic manifold (Abstract >)
The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations) and elastic selfinteractions (preferring ordered configurations). The elastic manifold model displays a depinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the highdimensional limit by Mézard and Parisi (1992) using the replica method.
We study the topology of the energy landscape of this model in the MézardParisi setting, computing the (annealed) topological complexity both of total critical points and of local minima. Our results confirm recent formulas by Fyodorov and Le Doussal (2020) and allows to identify the boundary between simple and glassy phases. The argument relies on asymptotics of large random determinants beyond invariant random matrices, and the analysis of the associated Matrix Dyson Equations. This is based on joint work with Gérard Ben Arous and Ben McKenna.

Zoom (link in EMail) 
Program HS 2020
Date/Time  Speaker  Title  Location 

16 September 2020  Jung Kyu Canci Hochschule Luzern 
The use of statistics in some pricing problems and
mixture of extreme value distributions (Abstract >)
We will consider two problems, which are related to some projects with
F.HoffmanRoche. The first one concerns the application of Stochastic Differential
Equations (Ito’s Integrals) in pricing problems. The second is about mixtures of
extreme value distributions studied with techniques from algebraic statistics.

Spiegelgasse 1 Room 00.003 
30 September 2020 16:00  Antonio Auffinger Northwestern University 
Counting minima and saddles in highdimensional random landscapes. (Abstract >)
I will discuss the complexity of minima and saddles in two classic examples of random landscapes: the spherical pspin glass model and the model of a single particle in a random confining potential. In these models, the number of critical points diverge exponentially as the dimension of the ambient space grows. By combining KacRice’s formula and random matrix tools, we are able to estimate this exponential growth. I will also explain some consequences of these results to statistical physics and optimization theory.

Online via Zoom 
21 October 2020  Kaspar Rufibach Roche 
Use of multistate models to improve decisionmaking in clinical trials
(Abstract >)
In this talk, I will illustrate how critical decisions in drug development are typically based on a tiny fraction of the collected data only. As an example, in early development oncology clinical trials, the decision whether to move a molecule to Phase 3 is typically based on response proportions and duration of response in those that respond, while in Phase 3 the primary endpoint will be longterm endpoints such as progressionfree (PFS) or overall survival (OS). Effects on responsebased shortterm endpoints seldom translate in effects on these relevant endpoints. We propose to make decisions not based on intermediate endpoints, but on a prediction of the OS hazard ratio (HR) between data of the new molecule collected in the early phase trial and historical data of the control treatment. This HR prediction is using a multistate model based on the various disease states a patient may go through until death. This yields a gating strategy with improved operating characteristics compared to traditional decision rules in the context of early phase clinical trials. If time permits I will further discuss how the joint distribution of PFS and OS as a function of transition probabilities in a multistate model can be derived. No assumptions on copulae or latent event times are needed and the model is allowed to be nonMarkov. From the joint distribution, statistics of interest can then readily be computed. As an example, we provide closed formulas and statistical inference for Pearson’s correlation coefficient between PFS and OS. Our proposal complements existing approaches by providing methods of statistical inference while at the same time working within a much more parsimonious modelling framework. The main conclusion of this talk is that multistate models are a useful and underutilized tool in the analysis of clinical trial data.

Online via Zoom 
28 October 2020  Mats Stensrud EPFL 
New estimands for causal inference conditional on a posttreatment event
(Abstract >)
Many studies aim to assess treatment effects on outcomes in individuals characterized by status on a particular posttreatment variable. For example, we may be interested in the effect of cancer therapies on quality of life, and quality of life is only welldefined in the those individuals who are alive. Similarly, we may be interested in the effect of vaccines on postinfections outcomes, which are only of interest in those individuals who become infected. In these settings, a naive contrast of outcomes conditional on the posttreatment variable does not have a causal interpretation, even in a randomized experiment. Therefore the effect in the principal stratum of those who would have the same value of the posttreatment variable regardless of treatment, such as the survivor average causal effect, is often advocated for causal inference. Whereas this principal stratum effect is a well defined causal contrast, it cannot be identified without strong untestable assumptions, and its practical relevance is ambiguous because it is restricted to an unknown subpopulation of unknown size. Here we formulate alternative estimands, which allow us to define the conditional separable effects. We describe the causal interpretation of the conditional separable effects, e.g. in settings with truncation by death, and introduce three different estimators. As an illustration, we use data from a randomized clinical trial to estimate a conditional separable effect of chemotherapies on quality of life in patients with prostate cancer.

Online via Zoom 
4 November 2020  Ronen Eldan Weizmann Institute 
Localization of measures on the Boolean hypercube with applications to interacting particle systems
(Abstract >)
Given a probability measure measure on the discrete hypercube, we describe a method to derive sufficient conditions for the measure to admit a "pure state decomposition", roughly speaking, to be expressed as a mixture of a small number of "localized" measures. Such a decomposition theorem has applications to meanfield approximation of Gibbs measures and to random graphs. Our method is based on the construction of a stochastic process which samples from the measure.

Online via Zoom 
18 November 2020  Diego Alberici EPFL 
The multilayer SK model: about the annealed and replica symmetric regions
(Abstract >)
A generalization of the SherringtonKirkpatrick (SK) model will be introduced, where spin variables are arranged over K layers and only consecutive layers interact. From the Machine Learning point of view this model can be thought as the random version of a deep Boltzmann machine.
A lower bound for the quenched pressure will be shown in terms of K SKmodels.
Then suitable conditions on the layers temperatures and sizes will be provided in order to guarantee that the quenched pressure coincides with the annealed one or satisfies a replica symmetric bound.

Zoom (Link in Email) 
25 November 2020  Harprit Singh Imperial College 
TBA  TBA 
2 December 2020  Eliran Subag Wiezmann Institute 
Generalized TAP approach for spherical spin glasses
(Abstract >)
The celebrated ThoulessAndersonPalmer approach suggests a way to relate the free energy of a meanfield spin glass model to the solutions of certain selfconsistency equations for the local magnetizations. I will describe a new geometric approach to define free energy landscapes for general spherical mixed pspin models and derive from them a generalized TAP representation for the free energy. Time permitting, I will explain how these landscapes are related to the pure states decomposition, ultrametricity property, and optimization of fullRSB models.

Zoom 
9 December 2020  Ricardo Schiappa Tecnico Lisboa 
Resurgence, Matrices, and Strings
(Abstract >)
I will give a broad overview of techniques in resurgent analysis and transseries, as they apply in string theory and matrix models (and with some focus on Painlevé type equations).

Zoom (link in Email) 
16 December 2020  Davide Gabrieli University of L'Aquila 
Remarks on the Interpolation Method
(Abstract >)
I will discuss a generalization of the classic condition of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the L^2 metric structure of the Gaussian random variables. As an example of application I will deduce the existence of the thermodynamic limit for a GREM model with infinite branches for which the classic conditions of validity fail. I will underline the dependence of the density of quenched free energy just on the metric structure and discuss the models from a metric viewpoint. Joint work with Roberto Boccagna.

Zoom (link in EMail) 
FS 2020
Date/Time  Speaker  Title  Location 

19 February 2020  Debapratim Banerjee Indian Statistical Institute, Kolkata 
Fluctuation of the free energy of SherringtonKirkpatrick model with CurieWeiss interaction: the paramagnetic regime >  Spiegelgasse 1 Room 05.002 
We consider a spin system containing pure two spin SherringtonKirkpatrick
Hamiltonian with CurieWeiss interaction. The model where the spins are spherically
symmetric was considered by Baik and Lee and Baik et al. which shows a two
dimensional phase transition with respect to temperature and the coupling constant.
In this paper we prove a result analogous to Baik and Lee in the “paramagnetic
regime” when the spins are i.i.d. Rademacher. We prove the free energy in this case
is asymptotically Gaussian and can be approximated by a suitable linear spectral
statistics. Unlike the spherical symmetric case the free energy here can not be
written as a function of the eigenvalues of the corresponding interaction matrix.
The method in this paper relies on a dense subgraph conditioning technique
introduced by Banerjee . The proof of the approximation by the linear spectral
statistics part is close to Banerjee and Ma .



Kaspar Rufibach Roche 
Multistate models to improve decisionmaking in clinical trials >  Spiegelgasse 1 Room 05.002 
In this talk, I will illustrate how critical decisions in drug development are
typically based on a tiny fraction of the collected data only. As an example, in
early development oncology clinical trials, the decision whether to move a molecule
to Phase 3 is typically based on response proportions and duration of response in
those that respond, while in Phase 3 the primary endpoint will be longterm
endpoints such as progressionfree (PFS) or overall survival (OS). Effects on
responsebased shortterm endpoints seldom translate in effects on these relevant
endpoints. We propose to make decisions not based on intermediate endpoints, but on
a prediction of the OS hazard ratio (HR) between data of the new molecule collected
in the early phase trial and historical data of the control treatment. This HR
prediction is using a multistate model based on the various disease states a
patient may go through until death. This yields a gating strategy with improved
operating characteristics compared to traditional decision rules in the context of
early phase clinical trials. If time permits I will further discuss how the joint
distribution of PFS and OS as a function of transition probabilities in a
multistate model can be derived. No assumptions on copulae or latent event times
are needed and the model is allowed to be nonMarkov. From the joint distribution,
statistics of interest can then readily be computed. As an example, we provide
closed formulas and statistical inference for Pearson’s correlation coefficient
between PFS and OS. Our proposal complements existing approaches by providing
methods of statistical inference while at the same time working within a much more
parsimonious modelling framework. The main conclusion of this talk is that
multistate models are a useful and underutilized tool in the analysis of clinical
trial data.
This is joint work with Ulrich Beyer, Jan Beyersmann, Matthias Meller, David Dejardin, and Uli Burger. 

Thursday 26 March 2020 16:00  Justin Ko Toronto 
The Free Energy of Spherical Spin Glasses with Vector Spins >  Zoom 
Spherical spin glass models can be interpreted as the optimization of a random
polynomial on a sphere in high dimensions. Computing the free energy will give a
formula for the typical maximum value of this random process as the dimension goes
to infinity. These models have appeared in areas such as physics, statistics, and
computer science. In this talk, we will present a free energy formula for coupled
systems of spherical spin glasses. We will describe several of the main
mathematical ideas in the derivation of this formula such as ultrametricity and
synchronization.


1 April 2020 11:00 new date!  Elena Pulvirenti Bonn 
Metastability for the dilute CurieWeiss model with Glauber Dynamics >  Zoom 
We analyse the metastable behaviour of the dilute Curie–Weiss model subject to a
Glauber dynamics. The model is a random version of a meanfield Ising model, where
the coupling coefficients are replaced by i.i.d. random coefficients, e.g.
Bernoulli random variables with fixed parameter p. This model can be also viewed as
an Ising model on the Erdos–Renyi random graph with edge probability p. The system
is a Markov chain where spins flip according to a Metropolis dynamics at inverse
temperature \beta. We compute the average time the system takes to reach the stable
phase when it starts from a certain probability distribution on the metastable
state (called the lastexit biased distribution), in the regime where the system
size goes to infinity, the inverse temperature is larger than 1 and the magnetic
field is positive and small enough. We obtain asymptotic bounds on the probability
of the event that the mean metastable hitting time is approximated by that of the
Curie–Weiss model. The proof uses the potential theoretic approach to metastability
and concentration of measure inequalities.



Alun Thomas The University of Utah, Division of genetic epidemiology 
Conjugate methods for estimating decomposable graphical models. >  Alte Universität, Seminarraum 201 
The cliqueseparator factorization of the joint distribution of random variables in
decomposable graphical models allows for conjugate choices for the prior
distributions of both parameters and graphical structure. I will describe a Markov
chain Monte Carlo approach for graphical model estimation when the parameter prior
has the Hyper Markov property and the graph prior is Structural Markov. I will
focus on computational considerations and consider extensions to the Weak
Structural Markov case.


6 May 2020  Valentina Ros ENS Paris 
Geometrical properties of highdimensional random landscapes: distribution of minima, saddles and activated dynamics >  Probability & Stats seminar Zoom room 
Characterizing the geometrical properties of nonconvex, highdimensional
landscapes is a longstanding problem in glassy systems, which is now gaining
increasing centrality in the fields of inference, machine learning, ecology and
biology. Among these properties, an important role is played by the statistics of
the stationary points (local minima, maxima and saddles), which is a crucial
ingredient to understand how the landscape is explored and optimized with local
dynamics. In this talk I will summarize some recent progress done in the analysis
of a prototypical random landscape, the energy functional of the socalled
spherical pspin model or random Gaussian monomial. I will discuss how to extract
information on the connectivity of suboptimal minima in configuration space and on
the distribution of the energy barriers separating them, using techniques of random
matrix theory. I will then comment on the dynamical implications of these results
on the dynamics, focusing in particular on the activated regime.


20 May 2020  Christian Brennecke Harvard 
Free Energy of the Quantum SherringtonKirkpatrick SpinGlass Model with Transverse Field >  Probability & Stats seminar Zoom room 
In this talk I will present a variational formula for the thermodynamic limit of
the free energy of the SherringtonKirkpatrick (SK) spinglass model with
transverse field, a quantum generalization of the classical SK model. Through its
path integral representation, the quantum SK model can be translated to a classical
vectorspin model with the spins taking values in the space of $ \{ 1, 1 \}
$valued cadlag paths. I will explain how to approximate the model by a sequence of
classical finitedimensional vectorspin models which enables us to use results of
Panchenko to determine the free energy in the thermodynamic limit. The talk is
based on joint work with Arka Adhikari.


27 May 2020  Weikuo Chen University of Minnesota 
Universality of Approximate Message Passing Algorithms >  Probability & Stats seminar Zoom room 
Approximate Message Passing (AMP) algorithms are nonlinear power iteration methods
originally arising from the context of compressed sensing. In this talk, I will
introduce a Lipschitizian functional iteration, as a generalization of the AMP
algorithms, and discuss its universality in disorder. In addition, I will explain
how our results imply universality in a number of AMPs popularly adapted in
Bayesian inferences and optimizations in spin glasses.

HS 2019
Date/Time  Speaker  Title  Location 

30 Oktober 2019  AlainSol Sznitman ETH Zurich 
On bulk deviations for the local behavior of random interlacements >  Spiegelgasse 1 Room 00.003 
In this talk we will discuss some recent large deviation asymptotics concerning the
local behavior of random interlacements on \(\mathbb Z^d\), \(d\ge 3\). In
particular, we will describe the link with previous results concerning macroscopic
holes left inside a large box, by the the adequately thickened connected component
of the boundary of the box in the vacant sets of random interlacements.


13 November 2019  PierreFrancois Rodriguez Imperial College 
On the phase transition for levelset percolation of the Gaussian free field >  Spiegelgasse 1 Room 00.003 
We will discuss recent progress regarding the geometry of the Gaussian free field
in three and more dimensions. Our results, based on joint work with H.
DuminilCopin, S. Goswami and F. Severo, deal with the percolation problem
associated to levelsets of the Gaussian free field, as first investigated by J.
Lebowitz and H. Saleur in 1986. We will examine the equality of several natural
critical parameters related to this model.


27 November 2019  Alexis Prévost University of Köln 
Percolation for the Gaussian free field on the cable system >  Spiegelgasse 1 Room 00.003 
Among percolation models with longrange correlations, the Gaussian free field on
discrete graphs has received a lot of attention over the last few years, but not so
much on its continuous equivalent, the cable system. I will present a brief history
of the question, and explain why, for this model, the critical parameter is
surprisingly equal to 0, and is thus explicitly known, on a large class of graphs.
There are different proofs of this result, through uniqueness of the infinite
cluster, Russo formula, exploration martingale or random interlacements, but also
examples where the critical parameter is not equal to 0. Finally, I will give the
law for the capacity of the clusters of the level sets of the Gaussian free field
on the cable system.
Joint work with Alexander Drewitz and PierreFrançois Rodriguez. 

4 December 2019  Razvan Gurau CNRS 
Invitation to random tensors >  Spiegelgasse 1 Room 00.003 
I will give an introduction to random tensors and their applications. In particular
I will describe an universality result for invariant probability measures for
tensors: under generic scaling assumptions, for large tensor sizes any invariant
tensor measure approaches a Gaussian. I will then discuss the implications of this
result, as well as ways to avoid it.


11 December 2019  Daniele Tantari Scuola Normale Superiore 
Direct/Inverse Hopfield model and Restricted Boltzmann Machines >  Spiegelgasse 1 Room 00.003 
Meanfield methods fail to reconstruct the parameters of the model when the dataset
is clusterized. This situation is found at low temperatures because of the
emergence of multiple thermodynamic states. The paradigmatic Hopfield model is
considered in a teacherstudent scenario as a problem of unsupervised learning with
Restricted Boltzmann Machines (RBM). For different choices of the priors on units
and weights, the replica symmetric phase diagram of random RBM’s is analyzed and in
particular the paramagnetic phase boundary is presented as directly related to the
optimal size of the training set necessary for a good generalization. The
connection between the direct and inverse problem is pointed out by showing that
inference can be efficiently performed by suitably adapting both standard learning
techniques and standard approaches to the direct problem.


18 December 2019  Nicolas Macris EPFL 
Optimal errors and phase transitions in highdimensional generalised linear models >  Spiegelgasse 1 Room 00.003 
Highdimensional generalized linear models are basic building blocks of current
data analysis tools including multilayers neural networks. They arise in signal
processing, statistical inference, machine learning, communication theory, and
other fields. I will explain how to establish rigorously the intrinsic
informationtheoretic limitations of inference and learning for a class of randomly
generated instances of generalized linear models, thus closing several old
conjectures. Examples will be shown where one can delimit regions of parameters for
which the optimal error rates are efficiently achievable with currently known
algorithms. I will discuss how the proof technique, based on the recently developed
adaptive interpolation method, is able to deal with the output nonlinearity and
also to some extent with nonseparable input distributions.

FS 2019
Date/Time  Speaker  Title  Location 

27 Febuary 2019  Mo Dick Wong University of Cambridge 
Universal tail profile of Gaussian multiplicative chaos >  Spiegelgasse 5 Room 05.002 
We study the tail probability of the mass of Gaussian multiplicative chaos and
establish a formula for the leading order asymptotics under very mild assumptions,
resolving a recent conjecture of Rhodes and Vargas. The leading order coefficient
can be described by the product of two constants, one capturing the dependence on
the test set and any nonstationarity and the other one encoding the universal
properties of multiplicative chaos. This may be seen as a first step in
understanding the full distributional properties of Gaussian multiplicative chaos.


20 March 2019  David Belius University of Basel 
Theory of Deep Learning 1: Introduction to the main
questions > slides 
Spiegelgasse 5 Room 05.002 
This is the first talk in a five part series of talks on deep learning from a
theoretical point of view, held jointly between the probability theory and machine
learning groups of the Department of Mathematics and Computer Science. The four
invited speakers that follow after this talk are young researchers who are
contributing in different ways to what will hopefully eventually be a comprehensive
theory of deep neural networks.
In this first talk I will introduce the main theoretical questions about deep neural networks: 1. Representation  what can deep neural networks represent? 2. Optimization  why and under what circumstances can we successfully train neural networks? 3. Generalization  why do deep neural networks often generalize well, despite huge capacity? As a preface I will review the basic models and algorithms (Neural Networks, (stochastic) gradient descent, ...) and some important concepts from machine learning (capacity, overfitting/underfitting, generalization, ...). 

27 March 2019  Levent Sagun EPFL 
Theory of Deep Learning 2: Overparametrization in neural
networks: an overview and a definition > slides 
Spiegelgasse 5 Room 05.002 
An excursion around the ideas for why the stochastic gradient descent algorithm
works well on training deep neural networks leads to considerations about the
underlying geometry of the related loss function. Recently, we gained a lot of
insight into how tuning SGD leads to better or worse generalization properties on a
given model and task. Furthermore, we have a reasonably large set of observations
that lead to the conclusion that more parameters typically lead to better
accuracies as long as the training process is not hampered. In this talk, I will
speculatively argue that as long as the model is overparameterized (OP), all
solutions are equivalent up to finite size fluctuations.
We will start by reviewing some of the recent literature on the geometry of the loss function, and how SGD navigates the landscape in the OP regime. Then we will see how to define OP by finding a sharp transition described by the models fitting abilities to its training set. Finally, we will discuss how this critical threshold is connected to the generalization properties of the model, and argue that life beyond this threshold is (more or less) as good as it gets. 

3 April 2019  Arthur Jacot EPFL 
Theory of Deep Learning 3: Neural Tangent Kernel:
Convergence and Generalization of Deep Neural Networks > slides 
Spiegelgasse 5 Room 05.002 
We show that the behaviour of a Deep Neural Network (DNN) during gradient descent
is described by a new kernel: the Neural Tangent Kernel (NTK). More precisely, as
the parameters are trained using gradient descent, the network function (which maps
the network inputs to the network outputs) follows a socalled kernel gradient
descent w.r.t. the NTK. We prove that as the network layers get wider and wider,
the NTK converges to a deterministic limit at initialization, which stays constant
during training. This implies in particular that if the NTK is positive definite,
the network function converges to a global minimum. The NTK also describes how DNNs
generalise outside the training set: for a least squares cost, the network function
converges in expectation to the NTK kernel ridgeless regression, explaining how
DNNs generalise in the socalled overparametrized regime, which is at the heart of
most recent developments in deep learning.


10 April 2019  Lenaïc Chizat Université ParisSud 
Theory of Deep Learning 4: Training Neural Networks in
the Lazy and Mean Field Regimes > slides 
Spiegelgasse 5 Room 05.002 
The current successes achieved by neural networks are mostly driven by experimental
exploration of various architectures, pipelines, and hyperparameters, motivated by
intuition rather than precise theories. Focusing on the optimization/training
aspect, we will see in this talk why pushing theory forward is challenging, but
also why it matters and key insights it may lead to. We will review some recent
results on the phenomenon of "lazy training", on the role of overparameterization,
and on training neural networks with a single hidden layer.


15 April 2019 (Monday) 13:00  Marylou Gabrié ENS 
Theory of Deep Learning 5: Information theoretic approach to deep learning theory: a test using statistical physics methods > slides  Spiegelgasse 5 Room 05.002 
The complexity of deep neural networks remains an obstacle to the understanding of
their great efficiency. Their generalisation ability, a priori counter intuitive,
is not yet fully accounted for. Recently an information theoretic approach was
proposed to investigate this question.
Relying on the heuristic replica method from statistical physics we present an estimator for entropies and mutual informations in models of deep model networks. Using this new tool, we test numerically the relation between generalisation and information. 

TBA


8 May 2019  Roland Bauerschmidt Universitiy of Cambridge 
The geometry of random walk isomorphisms >  Spiegelgasse 5 Room 05.002 
The classical random walk isomorphism theorems relate the local time of a random
walk to the square of a Gaussian free field. I will present nonGaussian versions
of these theorems, relating hyperbolic and hemispherical sigma models (and their
supersymmetric versions) to nonMarkovian random walks interacting through their
local time. Applications include a short proof of the SabotTarres limiting formula
for the vertexreinforced jump process (VRJP) and a MerminWagner theorem for
hyperbolic sigma models and the VRJP. This is joint work with Tyler Helmuth and
Andrew Swan.


15 May 2019  Augusto Teixeira IMPA 
Random walk on a simple exclusion process >  Spiegelgasse 5 Room 05.002 
In this talk we will study the asymptotic behavior of a random walk that evolves on
top of a simple symmetric exclusion process. This nice example of a random walk on
a dynamical random environment presents its own challenges due to the slow mixing
properties of the underlying medium. We will discuss a law of large numbers that
has been proved recently for this random walk. Interestingly, we can only prove
this law of large numbers for all but two exceptional densities of the exclusion
process. The main technique that we have employed is a multiscale renormalization
that has been derived from works in percolation theory.


Monday 17 June 2019 11:00  Shuta Nakajima University of Nagoya 
Gaussian fluctuations in directed polymers >  Spiegelgasse 5 Room 05.001 
In this talk, we consider the discrete directed polymer model with i.i.d.
environment and we study the fluctuations of the partition function. It was proven
by Comets and Liu that for sufficiently high temperature, the fluctuations converge
in distribution towards the product of the limiting partition function and an
independent Gaussian random variable. We extend the result to the whole L^2region,
which is predicted to be the maximal hightemperature region where the Gaussian
fluctuations should occur under the considered scaling. This is joint work with
Clément Cosco.

HS 2018
Date/Time  Speaker  Title  Location 

6 September 2018 
Lisa Hartung New York University 
The Ginibre ensemble and Gaussian multiplicative
chaos >
It was proven by Rider and Virag that the logarithm of the characteristic
polynomial of the Ginibre ensemble converges to a logarithmically correlated random
field. In this talk we will see how this connection can be established on the level
if powers of the characteristic polynomial by proving convergence to Gaussian
multiplicative chaos. We consider the range of powers in the \(L^2\) phase.
(Joint work in progress with Paul Bourgade and Guillaume Dubach). 
Spiegelgasse 1 Room 00.003 
19 September 2018 
Alexander
Drewitz Universität Köln 
Ubiquity of phases in some percolation models with
longrange correlations >
We consider two fundamental percolation models with longrange correlations: The
Gaussian free field and (the vacant set) of Random Interlacements. Both models have
been the subject of intensive research during the last years and decades, on
\(\mathbb Z^d\) as well as on some more general graphs. We investigate some
structural percolative properties around their critical parameters, in particular
the ubiquity of the infinite components of complementary phases.
This talk is based on joint works with A. Prévost (Köln) and P.F. Rodriguez (BuressurYvette). 
Spiegelgasse 1 Room 00.003 
31 October 2018 
Anton Klimovsky Universität DuisburgEssen 
Highdimensional Gaussian fields with isotropic
increments seen through spin glasses >
Finding the (spaceheight) distribution of the (local) extrema of highdimensional
strongly correlated random fields is a notorious hard problem with many
applications. Following Fyodorov and Sommers (2007), we focus on the Gaussian
fields with isotropic increments and take the viewpoint of statistical physics. By
exploiting various probabilistic symmetries, we rigorously derive the
FyodorovSommers formula for the logpartition function in the highdimensional
limit. The formula suggests a rich picture for the distribution of the local
extrema akin to the celebrated spherical SherringtonKirkpatrick model with mixed
pspin interactions.

Spiegelgasse 1 Room 00.003 
7 November 2018 
Dominik
Schröder IST Austria 
Cusp Universality for Wignertype Random Matrices
>
For Wignertype matrices, i.e. Hermitian random matrices with independent, not
necessarily identically distributed entries above the diagonal, we show that at any
cusp singularity of the limiting eigenvalue distribution the local eigenvalue
statistics are universal and form a Pearcey process. Since the density of states
typically exhibits only square root or cubic root cusp singularities, our work
complements previous results on the bulk and edge universality and it thus
completes the resolution of the WignerDysonMehta universality conjecture for the
last remaining universality type.

Spiegelgasse 1 Room 00.003 
14 November 2018 
Marius Schmidt Universität Basel 
Oriented first passage percolation on the
hypercube >
Consider the hypercube as a graph with vertex set \({0,1}^N\) and edges between two
vertices if they are only one coordinate flip apart. Choosing independent standard
exponentially distributed lengths for all edges and asking how long the shortest
directed paths from \((0,..,0)\) to \((1,..,1)\) is defines oriented first passage
percolation on the hypercube. We will discuss the conceptual steps needed to answer
this question to the precision of extremal process following the two paper series
"Oriented first passage percolation in the mean field limit" by Nicola Kistler,
Adrien Schertzer and Marius A. Schmidt: arXiv:1804.03117 [math.PR] and
arXiv:1808.04598 [math.PR].

Spiegelgasse 1 Room 00.003 
21 November 2018 
Antti Knowles University of Geneva 
Local law and eigenvector delocalization for
supercritical ErdosRenyi graphs >
We consider the adjacency matrix of the ErdosRenyi graph \(G(N,p)\) in the
supercritical regime \(pN > C \log N\) for some universal constant C. We show
that the eigenvalue density is with high probability well approximated by the
semicircle law on all spectral scales larger than the typical eigenvalue spacing.
We also show that all eigenvectors are completely delocalized with high
probability. Both results are optimal in the sense that they are known to be false
for \(pN < \log N\). A key ingredient of the proof is a new family of large
deviation estimates for multilinear forms of sparse vectors. Joint work with Yukun
He and Matteo Marcozzi.

Spiegelgasse 1 Room 00.003 
28 November 2018 
Gaultier
Lambert University of Zurich 
How much can the eigenvalue of a random matrix
fluctuate? >
The goal of this talk is to explain how much the eigenvalues of large Hermitian
random matrices deviate from certain deterministic locations. These are known as
“rigidity estimates” in the literature and they play a crucial role in the proof of
universality. I will review some of the current results on eigenvalues’
fluctuations and present a new approach which relies on the theory of Gaussian
Multiplicative Chaos and leads to optimal rigidity estimates for the Gaussian
Unitary Ensemble. I will also mention how it is also deduce a central limit theorem
from our proof. This is joint work with Tom Claeys, Benjamin Fahs and Christian
Webb.

Spiegelgasse 1 Room 00.003 
12 December 2018 
Ioan
Manulescu University of Fribourg 
Uniform Lipschitz functions on the triangular lattice
have logarithmic variations >
Uniform integervalued Lipschitz functions on a finite domain of the triangular
lattice are shown to have variations of logarithmic order in the radius of the
domain. The level lines of such functions form a loop \(O(2)\) model on the edges
of the hexagonal lattice with edgeweight one. An infinitevolume Gibbs measure for
the loop \(O(2)\) model is constructed as a thermodynamic limit and is shown to be
unique. It contains only finite loops and has properties indicative of
scaleinvariance: macroscopic loops appearing at every scale. The existence of the
infinitevolume measure carries over to height functions pinned at 0; the
uniqueness of the Gibbs measure does not. The proof is based on a representation of
the loop \(O(2)\) model via a pair of spin configurations that are shown to satisfy
the FKG inequality. We prove RSWtype estimates for a certain connectivity notion
in the aforementioned spin model. Based on joint work with Alexander Glazman.

Spiegelgasse 1 Room 00.003 