Columbia Probability Seminar

Friday, September 18, 2015

• Sumit Mukherjee (Columbia)

Limit Theorems in Graph Coloring Problems

Abstract:

In this talk the limiting distribution of the number of monochromatic edges in a uniform random coloring of any sequence of graphs will be characterized. The origin of these problems can be traced back to the classical birthday paradox, and are often helpful in the study of coincidences. I will show that the number of monochromatic edges converge in distribution to either a Poisson or a Normal when the number of colors grow to infinity. The results are universal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors, and work for any graph sequence, deterministic or random. The necessary and sufficient condition for asymptotic normality when the number of colors is fixed, will also be discussed. Finally, using results from the emerging theory of graph limits, the asymptotic distribution is also characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relates to results of Erdos and Alon extremal sub-graph counts.
This is based on joint work with Bhaswar Bhattacharya and Persi Diaconis.

Friday, September 25, 2015

• Li—Cheng Tsai (Stanford) [11 a.m. -- noon]

KPZ equation limit of interacting particle systems

Abstract:

In this talk we survey the weak universality (i.e. convergence to the Kardar-Parisi-Zhang (KPZ) equation) of interacting particle systems in the KPZ universality class. Such universality is considerable more accessible when a Hopf-Cole (HC) transformation linearizing the particle system exists. Here we derive such a HC transformation for the Higher Spin Exclusion Process (HSEP), introduced by Corwin and Petrov (2015), and hence for all known integrable models of the KPZ class in 1+1 dimensions. We further leverage this result into the weak universality of the HSEP. Next, we consider non-nearest neighbor exclusion processes, where an exact HC transformation is unavailable. We show that, for a subclass of models permitting certain gradient type conditions, an approximated form of the HC transformation exists, and prove the weak universality of the processes.
This talk is based on joint work with Ivan Corwin and joint work with Amir Dembo.

Slides

• Amir Dembo (Stanford) [noon -- 1 p.m.]

Walking within growing domains: recurrence versus transience

• Abstract:

  When is simple random walk on growing in time d-dimensional domains recurrent? For domain growth which is independent of the walk, we review recent progress and related universality conjectures about a sharp recurrence versus transience criterion in terms of the growth rate. We compare this with the question of recurrence/transience for time varying conductance models, where Gaussian heat kernel estimates and evolving sets play an important role.

  We also briefly contrast such expected universality with examples of the rich behavior encountered when monotone interaction enforces the growth as a result of visits by the walk to the current domain's boundary.

  This talk is based on joint works with Ruojun Huang, Ben Morris, Yuval Peres and Vladas Sidoravicius.

  Slides

Friday, October 2, 2015

• Minerva Lectures by Frank den Hollander

• Monday, September 28, 4-5:30pm, 507 Math

• Tuesday, September 29, 6-7:30pm, 507 Math

  Slides from Lectures 1 and 2

• Wednesday, September 30, 4:30-6pm, *520* Math

• Thursday, October 1, 5:30-7pm, 507 Math

• Friday, October 2, 11:30am-1pm, *520* Math

Friday, October 9, 2015, Courant Institute

• Columbia / Courant Joint Probability Seminar Series:

  
  
• Time: 9:30 am, Friday October 9th, Warren Weaver Hall, Room 512 (at Courant)


• 9:30-10:30 Hoi Nguyen, Anti concentration of random walks and eigenvalue repulsion of random matrices.
  
  Abstract: I will survey recent characterization results on random walks (in both abelian and non-abelian groups) which sticks to a small region unusually long. As an application, we demonstrate a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for Wigner matrices of discrete type.


• 10:30-11 Coffee break


• 11-12 Nina Snaith, Combining random matrix theory and number theory.
  
  Abstract: Many years have passed since the initial suggestion by Montgomery (1973) that in an appropriate asymptotic limit the zeros of the Riemann zeta function behave statistically like eigenvalues of random matrices, and the subsequent proposal of Katz and Sarnak (1999) that the same is true of families of more general L-functions. While this limiting behaviour is very informative, even more interesting are the intricacies involved in the approach to this limit, the understanding of which allows us to use random matrix theory in novel ways to shed light on major open questions in number theory.


• 12-1 Ramon van Handel, The norm of structured random matrices.
  
  Abstract: Understanding the spectral norm of random matrices is a problem of basic interest in several areas of pure and applied mathematics. While the spectral norm of classical random matrix models is well understood, existing methods almost always fail to be sharp in the presence of nontrivial structure. In this talk, I will discuss new bounds on the norm of random matrices with independent entries that are sharp under mild conditions. These bounds shed significant light on the nature of the problem, and make it possible to easily address otherwise nontrivial phenomena such as the phase transition of the spectral edge of random band matrices. I will also discuss some conjectures whose resolution would complete our understanding of the underlying probabilistic mechanisms.

Friday, October 16, 2015

• Minerva Lectures by Constantinos Kardaras

• Tuesday, October 13th, 4:10—5:25pm, 507 Math

• Friday, October 16th, 10:30-11:45am, 507 Math

• Tuesday, October 20th, 4:10—5:25pm, 507 Math

• Thursday, October 22nd, 4:10—5:25pm, 903 SSW

Friday, October 23, 2015

• Julien Reygner (Ecole des Ponts) [11 a.m. -- noon]

Probabilistic interpretation of conservation laws and optimal transport in one dimension

• Abstract:

  We consider partial differential equations that describe the conservation of one or several quantities, possibly taking an additional dissipation mechanism into account, set on the real line. Such models are for instance relevant in gas dynamics or in the study of road traffic. When the initial data of these conservation laws are monotonic and bounded, a probabilistic theory can be developed by interpreting the solutions as cumulative distribution functions on the line. The study of the associated stochastic processes and their approximations by interacting particle systems provides a Lagrangian description of the solution, that will be used to derive existence results together with numerical schemes, as well as estimates of stability and convergence to equilibrium or traveling waves.

  Slides

• Pierre Le Doussal (Ecole Normale supérieure and CNRS) [noon -- 1pm]

From replica bethe ansatz solution of KPZ growth to non-crossing directed polymers

Slides

Friday, October 30, 2015

• Yan Fyodorov (Queen Mary University of London)

A Nonlinear Analogue of May—Wigner Instability Transition

• Abstract:

  We study a system of N≫1 degrees of freedom individually relaxing with a rate μ and coupled via a smooth stationary random Gaussian vector field with both gradient and divergence-free components. We show that generically with increasing the ratio of the coupling strength to the individual relaxation rate the system experiences an abrupt transition from a topologically trivial phase portrait with a single stable equilibrium into topologically non-trivial regime characterized by exponential in N number of equilibria, vast majority of which is expected to be unstable. This picture is suggested to provide a nonlinear analogue of May-Wigner instability transition. The analysis invokes statistical properties of the elliptic ensemble of real asymmetric matrices, and raises interesting questions about real eigenvalues of such matrices. The presentation will be based on joint work with Boris Khoruzhenko arXiv:1509.05737.

   

November 2--11, 2015

• Minerva Lectures by Timo Seppalainen

• Monday, November 2nd, 4—5:30pm, 507 Math

  Sign up for dinner after the seminar

• Wednesday, November 4th, 5:30-7pm, 507 Math

• Monday, November 9th, 4—5:30pm, 507 Math

• Wednesday, November 11th, 5:30-7pm, 507 Math

Friday, November 6, 2015

• Timo Seppalainen (University of Wisconsin-Madison)

Large deviations for random paths in a random medium

• Abstract:

  We consider the asymptotics of the free energy of a class of models of random paths in a random medium. Special cases include the much-studied directed polymer in a random medium and random walk in random environment (RWRE). We prove the existence of the limiting free energy and obtain along the way two variational formulas for the limit. One formula minimizes over generalized gradients called cocycles, while the other one involves entropy and maximizes over invariant probability measures. As a corollary we get a fairly general quenched large deviation principle for RWRE. The behavior of these variational formulas is partly understood in some special cases. This is joint work with Firas Rassoul-Agha and Atilla Y{\i}lmaz.

Friday, November 13, 2015

• Alexander Moll (MIT)

Random partitions and the quantum Benjamin-Ono hierarchy

• Abstract:

  Jack measures on partitions occur naturally in the study of continuum circular log-gases in generic background potentials V at arbitrary values \beta of Dyson’s inverse temperature. Our main result is a law of large numbers (LLN) and central limit theorem (CLT) for Jack measures in the macroscopic scaling limit, which corresponds to the large N limit in the log-gas. Precisely, the emergent limit shape and macroscopic fluctuations of profiles of these random Young diagrams are the push-forwards along V of the uniform measure on the circle (LLN) and of the restriction to the circle of a Gaussian free field on the upper half-plane (CLT), respectively. At the critical inverse temperature \beta=2, this recovers Okounkov’s LLN for Schur measures (2003) and coincides with Breuer-Duits’ CLT for biorthogonal ensembles (2013).
  
  Our limit theorems follow from an all-order expansion (AOE) of joint cumulants of linear statistics, which has the same form as the all-order 1/N refined topological expansion for the log-gas on the line due to Chekhov-Eynard (2006) and Borot-Guionnet (2012). To prove our AOE, we rely on the Lax operator for the quantum Benjamin-Ono hierarchy with periodic profile V exhibited in collective field variables by Nazarov-Sklyanin (2013). The characterization of the limit laws as push-forwards follows from factorization formulas for resolvents of Toeplitz operators with symbol V due to Krein and Calderón-Spitzer-Widom (1958).

  Slides

Thursday, November 19, and Friday, November 20, 2015

• Northeast Probability Seminar

• The Fourteenth Northeast Probability Seminar (NEPS) will be held in Room 912/914 of the Kimmel Center at NYU on November 19th and Room 109 of the Courant Institute, NYU on November 20th, 2015.

  Full schedule here

• * Jian Ding (University of Chicago)
  "Some geometric aspects for two-dimensional Gaussian free fields."

• * Francis Comets (Université Paris Diderot)
  "Localization in one-dimensional log-gamma polymers"

• * Jean-François Le Gall (University Paris-Sud Orsay)
  "First-passage percolation on random planar graphs."

• * Mykhaylo Shkolnikov (Princeton University)
  "Edge of beta ensembles and the stochastic Airy semigroup."

Friday, November 27, 2015

• Thanksgiving

Friday, December 4, 2015

• Vu—Lan Nguyen (Universite Paris—Diderot)

Variants of geometric RSK, geometric PNG and the multipoint distribution of the log-gamma polymer

• Abstract:

  The geometric Robinson-Schensted-Knuth (gRSK) correspondence has played an important role in the recent analysis of the directed polymer model with a log-Gamma potential. In particular, it has led to the confirmation of the Tracy-Widom GUE asymptotic distribution of its point-to-point partition function. In a joint work with Nikos Zygouras, we reformulate the gRSK correspondence in a way that leads to explicit integral formulae for the joint distribution of the point-to-point partition functions at different space-time points.

Friday, December 11, 2015

• Leonid Koralov (University of Maryland)

Averaging, homogenization, and large deviation results for the study of randomly perturbed dynamical systems

• Abstract:

  In this talk we'll discuss several asymptotic problems that can be formulated in terms of PDEs and solved using probabilistic methods. The first set of problems concerns the asymptotic behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term. Here we employ an extension of the large-deviation theory of Freidlin and Wentzell. Another set of problems concerns equations with a small diffusion term, where the first-order term corresponds to an incompressible flow, possibly with a complicated structure of flow lines. Here we use an extension of the averaging principle. Finally, we'll consider equations with a small diffusion term with periodic coefficients in a large domain. Depending on the relation between the parameters, either averaging or homogenization need to be applied in order to describe the behavior of solutions. We'll discuss the transition regime. Different parts of the talk are based on joint results with D.Dolgopyat, M. Freidlin, M. Hairer, and Z. Pajor-Guylai.

---

 

Spring Semester 2015

Friday, January 30, 2015

• Louis-Pierre Arguin (CUNY / University of Montreal)

             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).

   

Friday, February 6, 2015

• David Kelly (NYU)

             Fast-slow systems with chaotic noise.

• Abstract:

  It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations. In this talk, we focus on fast-slow ODEs, where the fast, chaotic variables are fed into the slow variables to yield a diffusion approximation. In particular we focus on the case where the chaotic noise is multidimensional and multiplicative. The tools from rough path theory prove useful in this difficult setting.

   

Friday, February 13, 2015

• Riddhipratim Basu (UC Berkeley)

             Maximal Increasing Subsequence with a Defect Line and Lebowitz's Slow Bond Problem

• Abstract:

  For a Poisson process in the plane with intensity 1, the distribution of the maximum number of points on an oriented path from (0,0) to (N,N) has been studied in detail, culminating in Baik-Deift-Johansson's celebrated Tracy-Widom fluctuation result. We consider a variant of the model where one adds, on the diagonal, some additional points according to an independent one dimensional Poisson process with rate \lambda. The question of interest here is whether for all positive values of \lambda, this results in a change in the law of large numbers for the the number of points on a maximal path. A closely related question comes from a variant of Totally Asymmetric Simple Exclusion Process, introduced by Janowsky and Lebowitz. Consider a TASEP in 1-dimension, where the bond at the origin rings at a slower rate r<1. The question is whether for all values of r<1, the single slow bond produces a macroscopic change in the system. We provide affirmative answers to both the questions. Based on joint work with Vladas Sidoravicius and Allan Sly.

   

Friday, February 20, 2015

• Johannes Ruf (University College London)

             Convergence of local supermartingales and Novikov-type conditions for processes with jumps

• Abstract:

  In the first part of the talk, we characterize the event of convergence of a local supermartingale. Conditions are given in terms of its predictable characteristics and jump measure. Furthermore, it is shown that L^1-boundedness of a related process is necessary and sufficient for convergence. The notion of extended local integrability plays a key role. In the second part of the talk, we provide a novel proof for the sufficiency of Novikov-Kazamaki type conditions for the martingale property of nonnegative local martingales with jumps. The proof is based on explosion criteria for related processes under a possibly non-equivalent measure. This is joint work with Martin Larsson.

   

Friday, February 27, 2015, Columbia Mathematics Department, room 520, special starting time: 9:30am (breakfast at 9:00am in room 508)

• Columbia / Courant Joint Probability Seminar Series:

• Speakers: Vadim Gorin (MIT), Richard Kenyon (Brown), Greta Panova (U Penn)

              More information here.

Friday, March 6, 2015

• Columbia Princeton Probability Day (Princeton)

• Speakers: Davar Khoshnevisan (Utah), Fraydoun Rezakhanlou (Berkeley), Prasad Tetali (Georgia Tech), Balint Virag (Toronto), Alex Drewitz (Columbia), Leonid Petrov (Virginia)

              More information here.

March 9-11, 2015

• Minerva Lectures by Philippe Di Francesco
  
  Integrable Combinatorics
   

Friday, March 27, 2015

• Partha Dey (UIUC)

             High temperature limits for (1+1)-d directed polymer with heavy-tailed disorder.

• Abstract:

  The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel (2012). It was proved that at inverse temperature beta n^{-gamma} with gamma=1/4 the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of gamma. Based on joint work with Nikos Zygouras.

   

Friday, April 3, 2015

• Georg Menz (Stanford)

             A two scale proof of the Eyring-Kramers formula

• Abstract:

  We consider a drift-diffusion process on a smooth potential landscape with small noise. We give a new proof of the Eyring-Kramers formula which asymptotically characterizes the spectral gap of the generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The new proof exploits the idea that the process has two natural time-scales: a fast time-scale resulting from the fast convergence to a metastable state, and a slow time-scale resulting from exponentially long waiting times of jumps between metastable states. A nice feature of the argument is that it can be used to deduce an asymptotic formula for the log-Sobolev constant, which was previously unknown (joint work with Andre Schlichting).

   

 

April 8-10, 2015

• Minerva Lectures by Wendelin Werner

•  

Friday, April 10, 2015

• Wendelin Werner (ETH Zurich) [11 a.m. — noon]

             A simple graph-valued Markov processes and renormalization

• Abstract:

  We describe a simple graph-valued Markov process and will explain how it can provide an elementary approach to some renormalization group ideas and questions. In particular, a (small) perturbation of the conjectured continuous scaling limits of critical models appear as stationary distributions for these Markov chains. We also point out how to relate this approach to some of the known results concerning scaling limits for some two dimensional models (the relation to uniform spanning tree is joint work with Stéphane Benoist and Laure Dumaz).
  

   

• Boguslaw Zegarlinski (Imperial College London) [12 noon — 1 p.m.]

             Ergodicity and smoothing of Markov semigroups with Hoermander type generators in infinite dimensions

• Abstract:

  This will be about applications of techniques based on generalised gradient bounds

   

Friday, April 17, 2015

• James Nolen (Duke)

             Scaling limits for a random conductance problem

• Abstract:

  This talk is about a random conductance model. Imagine an electrical network defined on the integer lattice Z^d with random, iid conductances assigned to the links between nearest neighbors. The voltage at each node and the current across each link is determined by solving a linear system of equations which is analogous to solving a divergence-form elliptic PDE with random coefficients. An important object of study is known as the ``corrector", which describes the random fluctuations in voltage that result from a macroscopic voltage gradient across the network. I'll explain how, in dimension 3 or more, a rescaled version of this discrete random object converges to a Gaussian field with covariance structure that is similar in some ways to a Gaussian free field. I'll also describe a central limit theorem for the effective conductivity of a finite network as the size of the network grows. This includes joint work with Jean-Christophe Mourrat, and with Antoine Gloria.

   

April 20-24, 2015

• Minerva Lectures by Fabio Toninelli

            

Friday, April 24, 2015

• Fabio Toninelli (Lyon)

             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, and predicted stationarity of Gibbs measures.

•  
  

 

Friday, May 1, 2015

• Hugo Duminil-Copin (Geneva)

             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.

   

Wednesday, May 13, 2015

• Claudio Landim (IMPA Rio de Janeiro) [12 noon — 1 p.m., room: Math 507]

             Condensing zero-range processes

• Abstract:

  Zero-range processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. Consider a condensing zero-range process with $N$ particles evolving on a fixed and finite set $S$. In the diffusive time scale $N^2$ the fraction of particles at each site evolves as a diffusion whose drift is unbounded. The process remains absorbed at the boundary once it attains it, and it performs, after this hitting time, a diffusion on a lower dimensional simplex, similar to the original one, until all particles concentrate on a single site. In a longer time scale $N^{1+\alpha}$, $\alpha>1$, the site which concentrates all particles evolves as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk.

   

 

Fall Semester 2014

Friday, September 12, 2014

• David Belius (NYU)

             The subleading order of two dimensional cover times

• Abstract:

  The epsilon-cover time of the two dimensional torus by Brownian motion is the time it takes for the process to come within distance epsilon>0 from any point. Its leading order in the small epsilon-regime has been established by Dembo, Peres, Rosen and Zeitouni [Ann. of Math., 160 (2004)]. In this talk I will present a recent result identifying the second order correction. This correction term arises in an interesting way from strong correlations in the field of occupation times, and in particular from an approximate hierarchical structure in this field. Our method draws on ideas from the study of the extremes of branching Brownian motion. Joint work with Nicola Kistler.

   

Friday, September 19, 2014

• Roland Bauerschmidt (Harvard)

             Self-avoiding walks and spins in 4 dimensions

• Abstract:

  The weakly self-avoiding walk is a model of random polymers predicted to show the same universal behavior as the strictly self-avoiding walk. Similarly, the n-component |\varphi|^4 model is a ferromagnetic spin model predicted to have the same universal behavior as the Ising model for n=1, the rotor model for n=2, and the classical Heisenberg model for n=3. It is known that these apparently different models are closely related in the sense that self-avoiding walks can be seen as spins with "n=0" components. I will present results for both models in 4 dimensions. In particular, we show that the susceptibility has a logarithmic correction to the meanfield scaling behavior with exponent (n+2)/(n+8) for the logarithm, thus 1/4 for the weakly self-avoiding walk. For the specific heat, we show a similar fractional logarithmic divergence for 0<4, a double logarithmic divergence for n=4, and that it is bounded for n>4 components. This is joint work with David Brydges and Gordon Slade. The proofs rely on recently developed methods for renormalisation group analysis.

   

Friday, September 26, 2014

• Omer Angel (UBC Vancouver)

             Unimodular hyperbolic planar graphs

• Abstract:

  We study random hyperbolic planar graphs by using their circle packing embedding to connect their geometry to that of the hyperbolic plane. This leads to several results: Identification of the Poisson and geometric boundaries, a connection between hyperbolicity and a form of non-amenability, and a new proof of the Benjamini-Schramm recurrence result. Based on works with subsets of Martin Barlow, Ori Gurel-Gurevich, Tom Hutchcroft, Asaf Nachmias and Gourab Ray.

   

Friday, October 3, 2014

• Yuri Bakhtin (NYU)

             Burgers equation with random forcing

• Abstract:

  The Burgers equation is one of the basic nonlinear evolutionary PDEs. The study of ergodic properties of the Burgers equation with random forcing began in 1990's. The natural approach is based on the analysis of optimal paths in the random landscape generated by the random force potential. For a long time only compact cases of the Burgers dynamics on a circle or bounded interval were understood well. In this talk I will discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption on the random forcing. The main result is the description of the ergodic components and existence of a global attracting random solution in each component. The proof is based on ideas from the theory of first or last passage percolation. The kicked forcing case is an extension of the Poissonian forcing case considered in a joint work with Eric Cator and Kostya Khanin.

   

Friday, October 10, 2014, Courant Institute, Warren Weaver Hall, 512

• Columbia / Courant Joint Probability Seminar Series:

• Speakers: Yuri Bakhtin (Courant), Paul Bourgade (Courant), Ivan Corwin (Columbia) and Eyal Lubetzky (Courant)

              More information here.

Friday, October 17, 2014

• Andrey Sarantsev (U Washington)

             Collisions of competing Brownian particles

• Abstract:

  Consider a finite system of Brownian particles on the real line, with each particle having drift and diffusion coefficients depending on its current rank relative to other particles. A triple collision occurs when three or more particles occupy the same position at the same time. We find a necessary and sufficient condition for absence of triple collisions. We also investigate some other types of collisions. This continues the work by Ichiba, Karatzas and Shkolnikov.

   

Friday, October 24, 2014

• no seminar this week; there is, however, a colloquium by Stanislav Smirnov on Thursday, October 23 at 3:30 p.m. in Davis Auditorium

Friday, October 31, 2014

• Guillaume Barraquand (Paris 7) [11 a.m. — 12 noon]

             KPZ scaling theory for integrable exclusion processes

• Abstract:

  The KPZ scaling theory provides a general method to compute all model-dependent constants arising in limit theorems for a large class of exclusion processes. The validity of this heuristic approach is rigorously proved only for a few exactly solvable models. In this talk, we will discuss how the theory applies for q-deformed exclusion processes introduced by Borodin-Corwin and Povolotsky : The q-TASEP and the q-Hahn TASEP. We will also introduce a two-sided generalization of the q-Hahn TASEP that preserve the integrable structure and further confirm KPZ scaling theory. This is a joint work with Ivan Corwin.

   

• Michael Damron (U Indiana) [12 noon — 1 p.m.]

             Rate of convergence of the mean for sub-additive ergodic sequences

• Abstract:

  For a subadditive ergodic sequence {X_{m,n}}, Kingman's theorem gives convergence for the terms X_{0,n}/n to some non-random number g. In this talk, I will discuss the convergence rate of the mean EX_{0,n}/n to g. This rate turns out to be related to the size of the random fluctuations of X_{0,n}; that is, the variance of X_{0,n}, and the main theorems I will present give a lower bound on the convergence rate in terms of a variance exponent (if it exists). The main assumptions are that the sequence is not diffusive (the variance does not grow linearly) and that it has a weak dependence structure. Various examples, including first and last passage percolation, bin packing, and longest common subsequence fall into this class. This is joint work with Tuca Auffinger and Jack Hanson.

   

Friday, November 7, 2014

• Samuel Watson (MIT)

             A conformally invariant metric on CLE₄

• Abstract:

  We will discuss an exploration process, introduced by Wendelin Werner and Hao Wu, in which conformal loop ensemble (CLE_κ) loops grow uniformly from the boundary of a domain. We relate this process in the case κ = 4 to the set of level loops of the zero-boundary Gaussian free field, and we use this point of view to show that the exploration process is a deterministic function of the CLE loops. We describe how this gives rise to a conformally invariant metric on CLE₄, which we conjecture can be given a natural geometric interpretation. Based on joint work with Scott Sheffield and Hao Wu.

   

Friday, November 14, 2014

• Li-cheng Tsai (Stanford) [11 a.m. — 12 noon]

             Infinite Dimensional Stochastic Differential Equations for Dyson’s Brownian Motion

• Abstract:

  Dyson's Brownian Motion (DBM) describes the evolution of the spectra of certain random matrices, and is governed by a system of stochastic differential equations (SDEs) with a singular, long-range interaction. In this talk I will outline a construction of the strong solution of the infinite dimensional SDE that corresponds to the bulk limit of DBM. This is a pathwise construction that allows an explicit space with generic configurations. The ideas used further lead to a proof of the pathwise uniqueness of the solution and of the convergence of the finite to infinite dimensional SDE.

   

• Brian Rider (Temple) [12 noon — 1 a.m.]

             Matrix Dufresne Identities

• Abstract:

  The classical Dufresne identity relates the distribution of the infinite time integral of geometric Brownian motion (a quantity of relevance in, for example, mathematical finance) to an inverse gamma law. This fact, along with various extensions, lurks behind the scenes in the geometric lifting of Pitman’s 2M-X formula due to Matsumoto-Yor, as well as in O’Connell’s recent work on the quantum Toda lattice. I will describe what can be said thus far in a matrix setting, when the Wishart distribution replaces the gamma, and Brownian motion on the general linear group steps in for the one-dimensional geometric Brownian motion. This is joint work with B. Valkó (University of Wisconsin).

Thursday, November 20, and Friday, November 21, 2014

• Northeast Probability Seminar

Tuesday, December 2, 2014 (joint probability / applied math seminar)

• Tom Trogdon (Courant) [2:45 p.m. — 3:45 a.m., 214 Mudd Building]

             Universality in numerical computations with random data

• Abstract:

  How do numerical algorithms perform statistically? To begin to analyze this broad question we follow Pfrang, Deift and Menon (2012) and diagonalize random matrices with integrable systems. Universal fluctuations are observed in the time to first deflation. We contrast this empirical study with analytical results for naive eigenvalue algorithms. We also consider the conjugate gradient, GMRES and genetic algorithms. Universality in iteration count is observed and new insights into random matrix theory are discussed. Finally, we see real-world applications of these ideas in a model of neural computation and in data analysis. This is joint work with P. Deift, G. Menon and S. Olver.

Friday, December 5, 2014

• Wolfgang König (WIAS / TU Berlin)

             A variational formula for the free energy of an interacting many-body system

• Abstract:

  We consider N bosons in a box in the d-dimensional space in a large box under the influence of a mutually repellent pair potential. The particle density is kept fixed and positive. Our main result is the identification of the limiting free energy at positive, sufficiently high temperature in terms of an explicit variational formula. The thermodynamic equilibrium is described by the symmetrised trace of the negative exponential of the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of N interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviation analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. Our formula is not able to express the unboundedly long cycles; as a result we derive only lower and upper bounds. Our results and their shortcomes are at the heart of Bose-Einstein condensation.

   

Monday, December 15, and Tuesday, December 16, 2014 (Joseph Fels Ritt Lectures)

• Alain-Sol Sznitman (ETH Zurich) [4:30 p.m., Davis Auditorium, 412 Schapiro Center, CEPSR 530 West 120th Street (between Broadway and Amsterdam)]

             Disconnection, interlacements, and the Gaussian free field

• Click here for further information

 

Spring Semester 2014

Friday, Jan 24, 2014

Rough paths theory and Gaussian processes

Abstract [pdf]

• Sebastian Riedel (TU Berlin)

Friday, Jan 31, 2014

• Jürg Fröhlich (ETH Zürich & IAS Princeton)

             Quantum Probability Theory and the Emergence of Facts in Quantum Theory

• Abstract:

  This talk represents an experiment in explaining quantum theory to mathematicians. First, it is explained on a simple example in which way quantum probability theory differs from classical probability theory. A dictionary between the two fields is sketched. It is then outlined what "quantum dynamical systems" are, and the key concepts of "information loss" and "entanglement" are introduced. Subsequently, the theory of von 'Neumann Measurements' is briefly described. The talk concludes with the theory of 'Non-Demolition Measurements' and, on some examples, will hopefully clarify how "purification" of states arises and how "facts" emerge in quantum mechanics.

   

Friday, Feb 7, 2014

• Jose Blanchet (Columbia)

             Exact Simulation of Multidimensional Reflected Brownian Motion

• Abstract:

  A number of algorithms have been produced recently to simulate exactly from one dimensional diffusions. These are remarkable because the transition density of a diffusion is typically not known in closed form. One uses Lamperti's transformation to induce a constant diffusion coefficient. Then, a key step is to propose a sample from Brownian motion. The likelihood ratio can often be localized and the so-called acceptance step can be executed using a Poisson thinning procedure. While similar strategies can be used for one dimensional reflected diffusions, Multidimensional reflected Brownian motion (RBM) calls for a new methodology. This is mainly because there is no simulatable process that can be used as a proposal and the transition density of multidimensional RBM is not known. In this talk we explain this new methodology. (This talk is based on joint work with Chris Dolan and Karthyek Murthy).

   

Friday, Feb 14, 2014

 

• Manuel Cabezas (IMPA Rio de Janeiro) [11 a.m. — 12 noon]
   
• Site-recurrence for two-type annihilating random walks
  
  Abstract: We consider a particle system where particles can be of two types, A or B. The particles perform independent random walks on $Z^d$ until they meet a particle of opposite type. At that time both particles annihilate and are no longer present in the system. Particles of the same type do not interact. We assume that the initial configuration is independent and identically distributed over $x$ in $Z^d$. We prove that the system is site-recurrent, that is, we prove that, almost surely, the origin is visited infinitely often by particles. (joint work with L. Rolla And V. Sidoravicius)

 

• Georg Menz (Stanford) CANCELLED
   
• A two scale proof of the Eyring-Kramers formula
  
  Abstract: We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients : The first one shows that the Gibbs measures restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of the mean-difference by a new weighted transportation distance. It contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also allows to derive sharp estimates on the log-Sobolev constant. (joint work with A. Schlichting)

Friday, Feb 21, 2014

• Jiří Černý (U Vienna)

             Vacant set of random walk on discrete torus

• Abstract:

  We prove a phase transition in the behaviour of certain macroscopic observable on the vacant set of random walk on the d-dimensional discrete torus, d>2. To this end we discuss a technique of coupling of Markov chains so that their ranges almost coincide all the time (joint work with A. Teixeira).

   

Friday, Feb 28, 2014

• no seminar due to Minerva lectures by M. Hairer (see flyer).

Friday, Mar 7, 2014

• Rodrigo Bañuelos (Purdue) CANCELLED

             tba

• Abstract:

  tba

   

Friday, Mar 14, 2014

Uniformity of the late points of random walk on \Z^d_n for d \ge 3

Abstract [pdf]

• Perla Sousi (U Cambridge) [11 a.m. — 12 noon]

• Michele Salvi (TU Munich) [12 noon — 1 p.m.]

             Homogenization in the Random Conductance Model: A CLT for the Effective Conductance

• Abstract: In nature, most materials have a rather complicated microscopic structure. Nevertheless, the global properties of the material (such as heat or electric conduction) can be described by differential equations with smooth coefficients. An explaination to this behavior is offered by Homogenization Theory: The rapid oscillations of local coefficients caused by impurities average out, or homogenize, at a macroscopic scale. The Effective Conductance of random electrical networks, i.e. the minimum of the Dirichlet Energy, is a prominent example of a homogenizing quantity. When scaled by the volume of the domain, the Effective Conductance was already known in the 80s to have an almost sure constant limit, but the order of fluctuations has remained an open problem for almost thirty years. In our work, we were able to derive a (non-degenerate) Central Limit Theorem for this quantity, albeit under rather restrictive hypothesis (i.i.d. conductances on the network with small ellipticity contrast, square domains and linear boundary conditions). The proof is based on the corrector method and the Martingale Central Limit Theorem, while a key integrability condition is furnished by the Meyers estimate. (joint work with Marek Biskup and Tilman Wolff)

   

Friday, Mar 21, 2014

• no seminar (Spring break)

Friday, Mar 28, 2014

• Frank Aurzada (TU Darmstadt)

             Persistence probabilities

• Abstract:

  Persistence probabilities concern the question of the probability that a stochastic process has a long excursion. For example one asks for the rate of P( \sup_{0\leq t\leq T} X_t \leq 1 ) as T tends to infinity. Further, one studies the behavior of the process conditioned on having such an excursion. The talk will survey some recent results in this area, in particular for fractionally integrated processes and for fractional Brownian motion.

   

Friday, Apr 4, 2014

• Soumik Pal (U Washington)

             The geometry of relative arbitrage

• Abstract:

  Suppose we do not impose stochastic models on how stock prices will evolve in the future. Is it possible, by active trading, to do better than a market index (say, S&P 500)? In this second part of a two-part talk^* (which can nevertheless be understood independently) we will prove the following surprising fact in complete generality. If we restrict ourselves to portfolios that are functions of the current stock prices, there is exactly one class of trading strategies that achieves this goal. Remarkably, these strategies are produced as solutions of Monge-Kantorovich optimal transport problem on the multidimensional unit simplex with a cost function that can be described as the log partition function. The ideas are an interplay between convex analysis, probability theory, and the geometry of the unit simplex. Based on joint work with Leonard Wong.
  
  ^* See here for the first part of the talk.

   

Friday, Apr 11, 2014

• Columbia Princeton Probability Day

• More information [html]

Friday, Apr 18, 2014

• Laurent Saloff-Coste (Cornell)

             Edge traffic on graphs and random walks on free solvable groups

• Abstract:

  What is the probability that a simple random walk on the square grid return to its starting point at time n after traversing each edge an equal number of time in both direction? The step 2 free solvable group with r generators is the quotient of the free group on r generators by its second commutator subgroup. What is the probability that the simple random walk on this group is back to its starting point at time n? And how are these questions related to each other? In discussing these questions, I will avoid technical details and focus on the description of the problems and results and some of the main ideas. (joint work with Tianyi Zheng, Stanford)
   

   

Apr 21-25, 2014

• Minerva lectures by A. Borodin (see flyer).

Friday, Apr 25, 2014

• Jean Bertoin (U Zurich)

             Asymmetric Cauchy distribution and the destruction of large random recursive trees

• Abstract:

  The probability mass function $1/j(j+1)$ for $j\geq 1$ belongs to the domain of attraction of a completely asymmetric Cauchy distribution. The purpose of the talk is to review some of applications of this simple observation to limit theorems related to the destruction of random recursive trees.
  Specifically, a random recursive tree of size $n+1$ is a tree chosen uniformly at random amongst the $n!$ trees on the set of vertices $\{0,1, …, n\}$ such that the sequence of vertices along any segment starting from the root $0$ increases. One destroys this tree by removing its edges one after the other in a uniform random order. It was first observed by Iksanov and M\"ohle that the central limit theorems for the random walk with step distribution given above explains the fluctuations of the number of cuts needed to isolate the root. We shall discuss further results in the same vein.

   

Friday, May 16, 2014 (both talks take place in Mathematics, room 520)

• Siva Athreya (ISI Bangalore) [11 a.m. — 12 noon]

             Invariance principle for variable speed random walks on trees

• Abstract:

  We consider stochastic processes on $(T,r)$ a complete, locally compact tree-like metric spaces on their "natural scale" with boundedly finite speed measure $\nu$. If $(T,r)$ is a discrete tree, then the process is a continuous time nearest neighbor random walk which jumps at rate depending on $\nu$. If $(T,r)$ is path-connected, then the process has continuous paths and equals the so called $\nu$-Brownian motion on $(T,r)$. In this talk we will discuss an invariance principle which shows that speed-$\nu_n$ motions on $(T_n,r_n)$ converge weakly in path space to the speed-$\nu$ motion on $(T,r)$ provided that the underlying triples of metric measure spaces converge in the Gromov-Hausdorff-vague topology. This is joint work with Anita Winter and Wolfgang Lohr.

• Rongfeng Sun (NUS Singapore) [12 noon - 1 p.m.]

             Polynomial chaos and scaling limits of disordered systems

• Abstract:

  Inspired by recent work of Alberts, Khanin and Quastel, 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 an explicit Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide 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, 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 that warrant further studies. Joint work with F. Caravenna and N. Zygouras.

 

 

 

 

 

 

---

 

Fall Semester 2013

Friday, Sept 6, 2013

• Quentin Berger (USC)

             The critical curve of the Copolymer Model at weak coupling

• Abstract:

  We will present the Copolymer Model introduced by Sinai, used to describe a
  heteregeneous polymer chain (say composed of hydrophilic or lipophilic
  monomers) lying at the interface between two solvents (say water and oil).
  When increasing the temperature, one observes a localization/delocalization
  phase transition: at low temperature, the polymer stays close to the
  interface to maximize the matches, whereas at high temperature, it wanders
  in one of the two solvents.
  
  We focus here on the critical curve, that separates the two regimes in this
  phase transition. In particular, we study its weak coupling limit, which is
  known to be universal. The value of the limit has however been subject to
  discussions, leading to contradictory conjectures. The result we will
  present is that we are actually able to compute the limit, under the
  assumption that the underlying return distribution has a power-law tail
  with finite mean.
  
  (joint work with F. Caravenna, J. Poisat, R. Sun, N. Zygouras)

   

Friday, Sept 13, 2013

• Ivan Corwin (Columbia, Clay Mathematics Institute, MIT)

              Spectral theory for the q-Boson particle system

• Abstract: We develop spectral theory for the generator of the q-Boson particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP, this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. Some limits of our results relate to directed polymers and the Kardar-Parisi-Zhang equation.

Friday, Sept 20, 2013

• Leif Döring (U Zurich)

              Identifying and Representing Self-Similar Processes

• Abstract: The systematic study of self-similar Markov processes goes back to the seminal article of John Lamperti from '72. For self-similar processes with positive trajectories a relation to Lévy processes is nowadays well-known whereas a general theory for real-valued self-similar process is still to be built. I will discuss the present knowledge and introduce an approach via singular SDEs.

Friday, Sept 27, 2013

• David Mason (Delaware)

              Laws of the Iterated Logarithm for Self-normalized Levy Processes at Zero

• Abstract:

Friday, Oct 4, 2013, 10am-1:30pm in room Math 520, Columbia University

• Columbia / Courant Joint Probability Seminar Series:

• Kardar-Parisi-Zhang Universality

              Speakers:

              Herbert Spohn (TU Munich), Jeremy Quastel (U. Toronto), and Leonid Petrov (Northeastern)

              More information here.

Friday, Oct 11, 2013

• Omer Tamuz (Microsoft/MIT)

              Product group rigidity using probability

• Abstract: In this talk we will explain how, using probabilistic constructions such as
  random walks on groups and their Poisson boundaries, one can study the
  algebraic properties of a group, as well as the properties of the
  probability spaces on which it acts.

• Joint work with Yair Hartman

Friday, Oct 18, 2013

• Gaetan Borot (Max Planck Institute)

              All-order asymptotics in beta ensembles in the multi-cut regime

• Abstract: The beta ensemble is a particular model consisting of N strongly correlated real random variables. For specific values of beta, it is be realized by the eigenvalues of a random hermitian matrix whose distribution is invariant by conjugation, and in this case the model is exactly solvable in terms of orthogonal polynomials, and provide solutions to the Toda chain equations. I will give present results on the large N asymptotics up to O(N^{-infinity}) of the partition function, and the moments of the x_i's, away from critical points. I will give an idea about the methods we use, which do not not rely on techniques of exactly solvable models. Under fairly general assumptions, the x_i accumulate in the large N limit on a collection of (g + 1) segments of the real line. The asymptotic behavior depends much on g: if g = 0, there is a 1/N expansion, but if g > 0, the model features an oscillatory behavior at all orders in 1/N. As an application, we obtain the all-order asymptotics of orthogonal polynomials away from their zero locus, and of solutions of the Toda chain in the continuum limit.

  Based on joint work with A. Guionnet (MIT)

   

Friday, Oct 25, 2013

• Ofer Zeitouni (Weizmann Institute/Courant)

              Slowdown for branching Brownian motion

• Abstract: Branching Brownian motion is closely related to solutions of the
  KPP-Fisher equation; the distribution of the maximum is
  the solution at time $t$, started from an initial step function; it lags
  $c\log t$ behind a travelling wave solution. The constant $c$, which was
  first computed by Bramson using probabilistic methods, is related
  to the behavior of Brownian motion conditioned to remain above level $-1$
  up to time $t$. I will discuss a variant where the diffusivity of the
  Brownian motion changes macroscopically. The correction term exhibits then
  a phase transition, and in a cerrtain regime is of order $ct^{1/3}$, with
  the constant $c$ related to the first zero of Airy's function. The proof involves an
  obstacle problem for Brownian motion with curved boundary.
  (Joint work with Pascal Maillard   http://arxiv.org/abs/1307.3583)

Friday, Nov 1, 2013

• Omer Angel (UBC) — canceled

             

•  

Friday, Nov 8, 2013

• Hao Wu (MIT)

              On the conformally invariant growing mechanism in CLE4

• Abstract: CLE4 is the collection of level lines of GFF. Miller and Sheffield give the first coupling between GFF and CLE4 in the way that loops in the CLE4 are the outmost $\pm\lambda$-height level loop of the field. From the study of CLE4, Werner and me construct a time parameter for each loop in CLE4 such that the time parameter transforms in a conformally invariant way. From this construction of time parameter, Sheffield, Watson and me derive the second coupling between GFF and CLE4. In this talk, we will first give the background of SLE, CLE and GFF. And then, discuss the coupling between GFF and SLE4, the first and the second coupling between GFF and CLE4.

Friday, Nov 15, 2013

• John Imbrie (University of Virginia)

              Localization of eigenfunctions of random operators: A problem of percolation of resonances

• Abstract: I will review what is known and expected for the Anderson model--the lattice Schroedinger equation with a random potential. For strong disorder, eigenfunctions are localized, i.e. they decay exponentially in the distance from a point. I will discuss a new proof of this fact based on an iterative construction not unlike the Jacobi diagonalization procedure. Convergence depends on the diluteness of resonant regions in an associated correlated percolation problem. Similar issues arise in the problem of localization for many-body Hamiltonians with disorder. (Joint work with Tom Spencer).

Friday, Nov 22, 2013

• NEPS @ CUNY

              Hairer, Junge, Nikeghbali, Williams

• Further information

Friday, Nov 29, 2013

• Thanksgiving

•  

Friday, Dec 6 2013

 

• Lingjiong Zhu (Morgan Stanley) [11am-12noon]
• Title: Nonlinear Hawkes Processes
  
  Abstract: Hawkes process is a simple point process that has long memory, clustering effect, self-exciting property and is in general non-Markovian. The future evolution of a self-exciting point process is influenced by the timing of the past events. In this talk, we will first discuss the limit theorems for linear and nonlinear Hawkes processes. Then, we will drop the usual assumptions on Hawkes processes and categorize it into different regimes, i.e. sublinear, sub-critical, critical, super-critical and explosive regimes. We will discuss time asymptotics as well as some other properties.

 

• Nicola Kistler (CUNY) [12noon-1pm]
• Title: Derrida's Generalized Random Energy Models --- beyond spin glasses
  
  Abstract: The random energy models, REMs for short, are simple Gaussian fields with a built-in hierarchal structure. They have been introduced by Bernard Derrida in the 80's to shed light on the mysterious Parisi theory for mean field spin glasses. Within this context, the REMs have provided invaluable insights ever since. On the other hand, recent advances in disparate fields suggest that the REMs are truly universal objects which play a fundamental role well beyond the original spin glass setting. I will discuss the issue by means of a refinement of the second moment method which is remarkably efficient in the study of the extremes of large combinatorial structures whenever multiple scales can be identified.