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Extreme Value Statistics in Mathematics, Physics and Beyond
“Non-asymptotic bounds for the tail of the maximum of random fields”
Non asymptotic bounds for the tail of the maximum are useful in spatial statistics to ensure the level of a test. We present to methods for smooth Gaussian random fields
These two methods use in a different manner the Rice formula to compute the expectation of the number of maxima above a given level and the main difficulty is the computation of the expectation of the absolute value of the determinant Hessian matrix.
The first method is the record method that works on dimensions 2 and 3 and uses some Fourier method to perform this computation.
The second method is the direct method and uses random matrix theory.
“Last Passage Percolation on a compound Poisson process”
In this talk we will consider a compound Poisson process in the plane, and the longest up-right path between two points, where this path is allowed to pick up weights from the compound Poisson process. We will introduce the Busemann function for
this system, which is connected to semi-infinite geodesics, and show that this Busemann function is intimately related to equilibrium measures of a related interacting fluid process, a generalization of the Hammersley-Aldous-Diaconis interacting particle process. Control of these equilibrium measures would allow us to prove cube-root fluctuations which seem universal in these types of problems.
This is joined work with Leandro Pimentel
"Extreme value statistics of smooth random Gaussian fields and applications to cosmology"
I will first show how extreme value statistics can be relevant to cosmology, by focussing on the examples of the largest clusters in the observable Universe and of cold/hot spots in the cosmic microwave background temperature fluctuations. Then, analytical expressions for extreme value distributions will be derived by using counts-in-cells formalism, allowing us to write interesting approximations for the distribution of the extrema in patches of a 2D or 3D smooth Gaussian random field. These calculations will be performed in a rare event regime, but clustering effects will not, nethertheless, be neglected. Of course, many of the results discussed during my talk will be already known to mathematicians, and my practical point of view will often be not quite rigourous.
“Statistics at the tip of a branching random walk and simple models of evolution with selection”
The talk will present some recents results concerning the distribution of the rightmost particles of a branching brownian motion. It will also show the effect of the speed of evolution on the genealogy, for simple models of evolution with selection.
“Replica Bethe ansatz solution for one-dimensional directed polymers”
The distribution function of the free energy fluctuations in one-dimensional directed polymers with delta-correlated random potential is studied by mapping the replicated problem to the N-particle quantum boson system with attractive interactions. We describe the structure and the properties of the full set of eigenfunctions and eigenvalues of this many-body system and perform the summation over the entire spectrum of excited states. It is shown that in the thermodynamic limit the problem is reduced to the Fredholm determinant with the Airy kernel yielding the universal Tracy-Widom distribution, which is known to describe the statistical properties of the Gaussian unitary ensemble as well as many other statistical systems.
“Real-space Condensation and Extreme Value Statistics”
In this talk I shall review the phenomenon of real-space condensation wherein a finite fraction of interacting particles moving on a lattice condense onto a single site. The phenomenon can be understood by considering models with particularly simple
stationary states which have a factorised form. To this end I shall review necessary and sufficient conditions for factorised stationary states in a simple class of models. Then the condensation can be analysed and understood in terms of large deviations of sums of random variables and extreme value statistics.
“Correlations between record events in sequences of random variables with a linear trend”
The statistics of records in sequences of independent, identically distributed random variables is a classic subject of study. One of the earliest results concerns the stochastic independence of record events. Recently, records statistics beyond the case of i.i.d. random variables have received much attention, but the question of independence of record events has not been addressed systematically. In this paper, we study this question in detail for the case of independent, non-identically distributed random variables, specifically, for random variables with a linearly moving mean. We find a rich pattern of positive and negative correlations, and show how their asymptotics is determined by the universality classes of extreme value statistics. This classification is used to develop a test for heavy tails in data sets that works particularly well on small data sets on the order of a few dozen entries.
J. Franke, G. Wergen and J. Krug
“The sets of high excursions of Gaussian processes”
“The extremal process of branching brownian motion”
I will report on recent work with Louis-Pierre Arguin and Anton Bovier on the extremal process of branching Brownian motion. Our main result is an explicit construction of the limiting object in terms of a Poisson cluster process.
“High-dimensional Gaussian random fields with isotropic increments seen through spin glasses”
For a particle subjected to arbitrary Gaussian random potential with isotropic increments, we prove a computable saddle-point variational representation for the free energy in the infinite-dimensional limit. The proofs are based on the techniques developed in the course of the rigorous analysis of the Sherrington-Kirkpatrick model with vector spins.
"Eigenvalue order statistics for the heat equation with random potential"
We consider the random Schrödinger operator on the lattice with i.i.d. potential, which is double-exponentially distributed. In a large box, we look at the lowest eigenvalues, together with the location of the centering of the corresponding eigenfunction, and derive a Poisson process limit law, after suitable rescaling and shifting, towards an explicit Poisson point process. This is a strong form of Anderson localisation at the bottom of the spectrum. Since the potential is unbounded, also the eigenvalues are, and it turns out that the gaps between them are much larger than of inverse volume order. Our long-term goal is an application to concentration properties of the Cauchy problem, the parabolic Anderson model. This is joint work with Marek Biskup (Los Angeles and Budweis).
"Record statistics beyond the standard model"
The standard model of record statistics considers time series of independent, identically distributed (i.i.d.) random variables. In recent work, we and others have conducted case studies which extend the classic theory by relaxing the assumption of stationarity (in the sense of identical distributions) or independence, or both. The talk will begin by briefly reviewing the standard setting. I then describe some of our results for the linear drift model, in which a constant trend is added to an i.i.d. time series, and its application to record-breaking temperatures. Related results on the emergence of record correlations in the linear drift model, and on records in random walks with drift will be presented in subsequent talks by Jasper Franke and Gregor Wergen.
“Random Convex Hulls and Extreme Value Statistics”
Convex hull of a set of points in two dimensions roughly describes the shape of the set. In this talk, I will discuss the statistical properties of the convex hull of a set of N independent planar Brownian paths. I will show how to compute exactly the mean perimeter and the mean area of this convex hull, both for open and closed paths. The area and perimeter grow extremely slowly (logarithmically) with increasing population size N. This slow growth is a consequence of extreme value statistics and has interesting implications in ecological context of estimating the home range of a herd of animals with population size N.
"Maximal eigenvalue of a Gaussian random matrix: large deviations and Tracy-Widom"
In this talk, I will present some recent results for the distribution of the maximal eigenvalue of a Gaussian random matrix. For the Gaussian ensembles, it is known that the distribution of the maximal eigenvalue goes (in the limit of a large matrix) to the famous Tracy-Widom law. This law describes small typical fluctuations of the maximal eigenvalue around its mean value. One can also be interested in the far left or right tail of the distribution, these are described by the large deviations. I will explain how we could compute the right large deviation for the GUE case (Gaussian Unitary Ensemble) using a method of orthogonal polynomials. This method gives also a quite elementary rederivation of the Tracy-Widom law.
“Order statistics and applications to astrophysics”
Extreme value statistics (EVS) including order statistics is applied to the galaxy luminosities in the Sloan Digital Sky Survey. The DR6 Main Galaxy Sample is divided into red and blue subsamples and the Luminous Red Galaxy Sample is also treated separately. A non-parametric comparison of the extrem luminosities with the Fisher-Tippett-Gumbel distribution indicates good agreement provided uncertainties arising both from the finite size of the samples and from the sample size distribution are accounted for. The importance of correlations and of order statistics is discussed in connection with the Tremain-Richstone ratios for the gap between the luminosities of the brightest and second brightest galaxies in a cluster.
“Extremes in log-correlated variables”
In this talk I present some recent results on the extreme statistics of Gaussian Log-correlated variables. This problem has strong similarities with disordered systems displaying a freezing transition. Application to Burgers turbulence will be also discussed.
“Spontaneous Resonances and the Coherent States of Queuing Networks”
We present an example of a highly connected closed network of servers, where the time correlations do not vanish in the infinite volume limit. The limiting interacting particle system behaves in periodic manner. This phenomenon is similar to the continuous symmetry breaking at low temperatures in statistical mechanics, with the average load plaing the role of the inverse temperature. This talk based on joint paper with S.Shlosman and A.Vladimirov.
"Extreme statistics of vicious walkers: from random matrices to Yang-Mills theory"
Non-intersecting random walkers (or ''vicious walkers'') have been studied in various physical situations, ranging from polymer physics to wetting and melting transitions and more recently in connection with random matrix theory or stochastic growth processes. In this talk, I will present a method based on path integrals associated to free Fermions models to study such statistical systems. I will use this method to calculate exactly the cumulative distribution function (CDF) of the maximal height of p vicious walkers with a wall (excursions) and without a wall (bridges). In the case of excursions, I will show that the CDF is identical to the partition function of 2d Yang Mills theory on a sphere with the gauge group Sp(2p). Taking advantage of a large p analysis achieved in that context, I will show that the CDF, properly shifted and scaled, converges to the Tracy-Widom distribution for $\beta = 1$, which describes the fluctuations of the largest eigenvalue of Random Matrices in the Gaussian Orthogonal Ensemble.
“Cracks in 1+1 ballistic deposition and discrete shocks in Burgers–type equations”
We consider the growth of clusters in the standard (1+1)-dimensional ballistic deposition process in a bounding box. We identify the channels separating clusters with shocks in a discrete Burgers type equation. Using this analogy and scaling considerations, we compute critical exponents characterizing the process, such as 2/3 in the height dependence of the average number of clusters in a horizontal cross-section, and 7/5 in the mass distribution of clusters. We also propose a new kind of equipped Airy process for ballistic growth and analyze numerically its two-point correlation function.
Joint work with K. Khanin (Toronto), S. Nechaev (Paris/Moscow), G. Oshanin (Paris/Stuttgart), and O. Vasilyev (Stuttgart/Moscow).
Aernout van Enter
“Bootstrap percolation, the role of anisotropy”
Bootstrap percolation models describe growth processes, in which a critical droplet in a metastable situation nucleates. In the statistics of droplet sizes, only for extreme values nucleation will occur. The occurrence of such critical droplets is ruled by asymptotic probabilities. We discuss how the proper scaling is modified in the presence of anisotropy. This is based on joint work with Tim Hulshof, Hugo Duminil-Copin and Anne Fey.
"How many eigenvalues of a Gaussian matrix are positive?"
The index of a random matrix (i.e. the number of positive or negative eigenvalues) is a random variable providing information about the stability of stationary points in high-dimensional potential landscapes. For a Gaussian matrix model of large size N, typically half of the eigenvalues are positive and half negative (Wigner's semicircle law), however atypical fluctuations away from the semicircle are quite interesting for a number of reasons. Using a Coulomb gas technique and functional methods, we compute analytically the full large deviation function of the index and find that it has a quadratic form modulated by a logarithmic singularity around the 'peak'. The distribution of the index has thus a Gaussian form near the peak, but its variance increases logarithmically with the matrix size. This finding is compared with an exact finite N result based on the Andreiéf integration formula and a recent result by Prellberg.
“Record statistics for biased random walks, with an application to financial data”
In this talk I will discuss the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff and is well understood. After introducing the model I will present some of our new results for the record rate of a random walk with a Gaussian jump distribution that is shifted by a constant drift. In the second part of my talk I will compare these results to the record statistics of 366 daily stock prices from the Standard & Poors 500 index. The biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices, and after detrending the number of upper records is in good agreement with the symmetric random walk. However the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified.