PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"

TL;DR

This thesis explores extreme value statistics in strongly correlated systems, including trapped fermions, random matrices, and random walks, providing exact mappings, edge behavior descriptions, and universal fluctuation results.

Contribution

It introduces new precise descriptions of edge statistics in fermionic systems and solves open problems in extreme value theory for random matrices and random walks.

Findings

Exact mappings between trapped fermions and random matrix ensembles

Complete description of largest eigenvalue fluctuations in Ginibre ensemble

Analytical results for gap statistics in discrete random walks

Abstract

In this thesis, we study three physically relevant models of strongly correlated random variables: trapped fermions, random matrices and random walks. In the first part, we show several exact mappings between the ground state of a trapped Fermi gas and ensembles of random matrix theory. The Fermi gas is inhomogeneous in the trapping potential and in particular there is a finite edge beyond which its density vanishes. Going beyond standard semi-classical techniques (such as local density approximation), we develop a precise description of the spatial statistics close to the edge. This description holds for a large universality class of hard edge potentials. We apply these results to compute the statistics of the position of the fermion the farthest away from the centre of the trap, the number of fermions in a given domain (full counting statistics) and the related bipartite entanglement entropy. Our analysis also provides solutions to open problems of extreme value statistics in random matrix theory. We obtain for instance a complete description of the fluctuations of the largest eigenvalue in the complex Ginibre ensemble. In the second part of the thesis, we study extreme value questions for random walks. We consider the gap statistics, which requires to take explicitly into account the discreteness of the process. This question cannot be solved using the convergence of the process to its continuous counterpart, the Brownian motion. We obtain explicit analytical results for the gap statistics of the walk with a Laplace distribution of jumps and provide numerical evidence suggesting the universality of these results.

Full text

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Introduction

Predicting the occurrence of extreme events is a crucial issue in many contexts, ranging from meteorology (hurricanes, heatwave, floods), geology (earthquakes), finance (stock market crash), all the way to physics. These events, although atypical, usually have a disastrous impact. To better prepare for such dramatic effects, it is vital to have a deeper understanding of these phenomena. Extreme value statistics have been studied in detail for a number of years since the seminal work of Gumbel [gumbel2012statistics] who identified the three universality classes in the case of independent and identically distributed (i.i.d) random variables: (I) Gumbel, (II) Fréchet and (III) Weibull. Many studies have been conducted ever since to obtain extreme value results in the case of either non-identically distributed or correlated random variables (see e.g. [krapivsky2000traveling, tracy1994level, bertin2006generalized, carpentier2001glass, majumdar2014extreme, bouchaud1997universality, majumdar2005airy]) but there is no general theory and each case needs to be studied separately.

In the context of physics, the applications of extreme value theory are numerous. A seminal example is the study of disordered systems [derrida1981random, bouchaud1997universality, dean2001extreme, le2003exact, schawe2018ground] such as (spin) glasses where the physics at low temperature is dominated by the ground state properties, i.e. the state with lowest energy (see also [biroli2007extreme] for a review). Extreme value statistics have also been extensively studied in the context of growth processes in the $(1+1)d$ Kardar-Parisi-Zhang universality class [kardar1986dynamic, johansson2000shape, prahofer2000universal, sasamoto2010one, calabrese2010free, amir2011probability, baik2012joint, baik1999distribution] and the related problem of directed polymer (see [majumdar2007course, TAKEUCHI201877] for pedagogical introductions). The turning point has been to connect this class of problems to random matrix theory (RMT), where Tracy and Widom have obtained exact results [tracy1994level]. The Tracy-Widom distribution has become ever-since a cornerstone of extreme value statistics for correlated systems [majumdar2014top]. It has been observed experimentally in the growth of nematic liquid crystals [takeuchi2010universal, takeuchi2011growing] in direct relation to this class of models but also in a very different context in coupled optical fibres experiments [fridman2012measuring]. However, in most physically relevant problems, obtaining the extreme value statistics remains an open problem. One of the main goal of this thesis is to enlarge the knowledge of extreme value statistics for correlated systems by exploring models for which these statistics can be obtained exactly.

In this thesis, we consider three large classes of models, with direct physical applications, where the extreme value statistics can be obtained exactly: (i) non-interacting fermions, (ii) random matrices and (iii) random walks.

0.0.1 Non-interacting fermions

We first discuss some physical properties of trapped gas of spin-less (or spin-polarised) non-interacting fermions. Even in the absence of interaction, the Pauli exclusion principle introduces strong quantum correlations in the system as two fermions cannot occupy the same quantum state. Of course, thermal fluctuations wash out these correlations at higher temperature and the quantum correlations are therefore more prominent at low temperature. The quantum statistics yields non-trivial spatial properties for the system of fermions. Cold atom experiments are the ideal platform to study these correlations as the recent experimental progress allows an unprecedented control over the parameters (see [bloch2008many, giorgini2008theory] for reviews). In these systems, the interactions between atoms can be tuned via the Feshbach resonance [regal2003tuning] and in particular, the non-interacting regime is reachable, allowing to probe and single-out the purely quantum effects emerging form the Pauli exclusion principle. These experiments are now used as quantum simulators for condensed-matter systems, where one can tune the parameters of the Hamiltonian in a controlled manner [bloch2012quantum]. For Fermi gases the recent development of Fermi gas microscope [haller2015single, cheuk2015quantum, parsons2015site] allows to probe the positions of single particles as observed in Fig. 1 (see [kuhr2016quantum, ott2016single] for recent reviews). This type of imaging could also allow to test dynamical and non-equilibrium properties of quantum gases which has generated a lot of theoretical interest over the years [calabrese2006time, calabrese2007quantum, vignolo2001one, krapivsky2019return, krapivsky2018quantum, eisler2013full, perfetto2017ballistic].

To conduct these types of experiments, the quantum gas needs to be confined by a trapping potential. This confining potential always creates a finite edge beyond which the density of particles in the gas is essentially zero. Although standard techniques such as local density approximation [castin2006basic] were developed to describe the gas in the bulk of the density, they are not able to capture the spatial statistics close to the edge [kohn1998edge, vignolo2000exact]. In view of the aforementioned experimental set-ups, it becomes crucial to understand properly the physics of the cold Fermi gas close to the edge. This issue was tackled in a recent series of paper [dean2015universal, dean2016noninteracting, dean2015finite] for non-interacting fermions, using the framework of determinantal point processes [hough2006determinantal, johansson2005random]. In particular, a lot of progress ensued from a direct connection between the ground state of a one-dimensional system of fermions at zero temperature confined by a harmonic potentials and the Gaussian Unitary Ensemble (GUE). The ground state joint probability of the positions of this Fermi gas can be computed exactly \be—Ψ_0(x_1,⋯,x_N)—^2=1ZN(α) ∏_i¡j—x_i-x_j—^2 ∏_i=1^N e^-α^2 x_i^2  ,  α=mωℏ . \eeOne recognises under the exact mapping $x_{i}=\sqrt{N}\lambda_{i}/\alpha$ the famous joint distribution of the eigenvalues of the GUE. This matrix ensemble is an invariant ensemble of random matrix theory, where Hermitian matrices are built with complex Gaussian independent entries. The result in Eq. \eqrefjPDF_OH1d_intro only holds for a harmonic potential, but it is experimentally relevant to design different shapes for the confining potential [mukherjee2017homogeneous, hueck2018two]. In fact, it was shown that the results obtained for the correlations at the edge of the density extend for any smoothly varying potential, e.g. $V(x)\sim|x|^{p}$ with $p>0$. These potentials yield a smooth variation of the density close to the edge. In random matrix theory, this universality class for the edge behaviour is usually referred to as soft edge. However, there are many examples in RMT, e.g. the Wishart or Jacobi ensembles, where the density has hard edges where it vanishes abruptly due to the presence of effective hard walls. It is therefore natural to ask if these RMT models have any counterparts in models of fermions, which present a similar hard edge behaviour and if there is a universality class associated to this different behaviour. This is one of the main purposes of this thesis.

0.0.2 Random matrices

A related problem that we consider is the statistical properties of random matrices [mehta2004random, anderson2010introduction, tracy1993introduction, livan2018introduction, forrester2010log]. Since its first appearance in the statistical literature [wishart1928generalised], RMT has been used extensively in mathematics, telecommunication, ecology or finance. In physics, it was first introduced by Wigner [wigner1951statistical] to describe the level spacing between energies of nuclei but has been used since in statistical physics in the context of vicious (non-intersecting) random walkers [Forrester1989, tracy2007nonintersecting, NAGAO200329, nadal2009nonintersecting, PhysRevLett.101.150601, FORRESTER2011500], one component plasma [chafai2014note, forrester1998exact, cunden2016large, cunden2017universality], mesoscopic physics [Jayannavar1989, beenakker1997random, Grabsch2017_2, PhysRevLett.101.216809, PhysRevB.81.104202, PhysRevLett.82.4220, PhysRevLett.78.4737] or quantum chromodynamics [wadia1980n, gross1993possible] (see [majumdar2014top, biroli2007extreme] for reviews on the physical applications of RMT). The models of random matrices offer a very useful setting to analyse extreme value statistics: while their eigenvalues are strongly correlated, their extreme value statistics can be obtained exactly and have been studied in detail [tracy1994fredholm, tracy1994level, tracy1994level2, duenez2010lowest, dumitriu2008distributions, rider2003limit]. In this thesis, we explore the connection between fermions and random matrices [dean2015finite, dean2015universal, dean2018wigner, eisler2013universality, grabsch2018fluctuations, le2016exact, marino2016number, eisler2013full, calabrese2015random] and mainly consider the Jacobi Unitary Ensemble (JUE) and complex Ginibre Ensemble. From these connections, we solve several open questions for the large deviations of the full counting statistics (FCS) and extreme value statistics in these ensembles.

0.0.3 Random walks

The last system that we will consider is random walks and its associated continuous counterpart Brownian motions. This seminal model of statistical mechanics was introduced for the first time more than a century ago by Louis Bachelier in the context of finance [bachelier1900theorie] (see also [Pearson1905] for the first appearance of the name random walk and [RevModPhys.15.1] for a review). In this model, the positions taken by the random walker form a strongly correlated set of random variables and is therefore a useful laboratory to test the effects of strong correlations. In particular, many results were obtained for the statistics of the global maximum [feller1968introduction, Mounaix_2018, comtet2005precise, majumdar2010universal]. The order statistics i.e. the statistics of the ordered maxima (second, third, etc) were also considered in detail both for a regular random walk [wendel1960order, port1962elementary, feller1968introduction, schehr2012universal, schehr2014exact, dassios1996sample, 10.2307/2959757] and for branching Brownian motions [Brunet_2009, Derrida2011, ramola, ramola2]. Order statistics is part of the general fluctuation theory which has been extensively studied in the mathematics community [wendel1960order, port1962elementary, feller1968introduction, pitman2018guide, revuz2013continuous]. A large literature has also emerged on the related topic of records for these random walks [majumdar_ziff, PhysRevE.86.011119, Sabhapandit_2011, majumdar2012record, schehr2014exact, berkowitz_records_noise, wergen_borgner_krug, Godr_che_2017, Godr_che_2014]. In both cases, the problem is simplified in the large $n$ limit and for finite variance jump distribution by using the convergence of this process towards Brownian motion, for which a number of results have been obtained [yor1995distribution, perret2013near, ramola, dassios1995distribution]. There exists however fewer results for the gap statistics of random walks [Brunet_2009, Derrida2011, ramola, battilana2017gap, schehr2012universal], which is inherently linked to the discrete nature of the process. These problems cannot be solved using the convergence to Brownian motion. A goal of this thesis is therefore to obtain new results for this interesting and versatile problem.

Overview of the thesis and main results

We present a quick overview of the thesis and summary of the main results. These main results are framed ($\boxed{x}$) in the text and the unpublished results are doubly framed ($\boxed{\framebox(0.0,0.0)[bl]{\boxed{x}}}$).

While some of the considered models are quite specific, many of their properties exhibit universality, as in many instances in statistical mechanics, and hold in a more general context.

First part: Spatial description of non-interacting fermions

Part LABEL:Part:Fer of this thesis is devoted to the study of non-interacting fermions and their connections to eigenvalues of random matrices.

In chapter LABEL:chap:fermions_intro, we introduce the framework to describe the spatial properties of fermions and review the results for smooth confining potentials.

In chapter LABEL:ch:_ferm_hard_edge, we extend the description of the edge statistics of non-interacting fermions to hard edge potentials. We show an exact mapping between the ground state of fermions trapped in a one-dimensional hard box potential and the Jacobi Unitary Ensemble of random matrices. We obtain exact results for the correlation kernel associated to this determinantal point process and show that these results extend to a new class of hard edge potentials. We extend these results in two directions: in higher dimension and at finite temperature. We apply these results to compute the fluctuations of the position of the fermion the farthest away from the centre of the trapping potential. In particular, we obtain the emergence of an intermediate deviation regime connecting the typical fluctuations to the large deviations, which does not appear for standard invariant ensembles as the GUE.

The study of these non-interacting fermions in hard edges led to the publication of two articles:

LABEL:Art:fermions_lett Statistics of fermions in a d-dimensional box near a hard wall

B. Lacroix-A-Chez-Toine, P. Le Doussal, S. N. Majumdar, G. Schehr,

Europhys. Lett. 120 (1), 10006 (2018).

LABEL:Art:ferm_long Non-interacting fermions in hard-edge potentials,

B. Lacroix-A-Chez-Toine, P. Le Doussal, S. N. Majumdar, G. Schehr,

J. Stat. Mech 12, 123103 (2018).

In chapter LABEL:ch:_rot_trap, we unveil an exact mapping between the ground state of a model of non-interacting fermions in rotation and the complex Ginibre ensemble. We compute exactly for this system the full counting statistics and entanglement entropy for any finite number $N$ of fermions. This problem is mapped to a specific case of the two-dimensional one component plasma, where charged particles (interacting via the long-ranged $2d$ Coulomb logarithmic repulsion) are confined by a harmonic potential. Extending to more generic potentials, we show the universality of the results for the full counting statistics in the plasma model. Revealing the emergence of intermediate deviation regimes for (i) the fluctuations of the particle the farthest away from the centre of the trap and (ii) the full counting statistics, we solve two puzzles of matching between the typical fluctuations and the large deviations.

The study of these non-interacting fermions in rotating traps, their connection to the complex Ginibre ensemble and their extension to general one component plasma led to the publication of two articles:

LABEL:Art:rot Rotating trapped fermions in two dimensions and the complex Ginibre ensemble: Exact results for the entanglement entropy and number variance,

B. Lacroix-A-Chez-Toine, S. N. Majumdar, G. Schehr,

Phys. Rev. A 99 (2), 021602 (2019).

LABEL:Art:r_max *Extremes of $2d$ Coulomb gas: universal intermediate deviation regime

*B. Lacroix-A-Chez-Toine, A. Grabsch, S. N. Majumdar, G. Schehr,

J. Stat. Mech 1, 013203 (2018).

We also recently submitted for peer-review another article on this subject:

LABEL:Art:FCSGin *Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

*B. Lacroix-A-Chez-Toine, J. A. Monroy Garzón, C. S. Hidalgo Calva, I. Pérez Castillo, A. Kundu, S. N. Majumdar, G. Schehr

arXiv preprint, arXiv: 1904.01813, (2019).

Second part: Statistics of the gaps of random walks

Part LABEL:Part:gaps of this thesis is devoted to the study of extreme value, order and gap statistics of random walks.

In chapter LABEL:intro_iid, we review a few results for the order and gap statistics of i.i.d. random variables.

In chapter LABEL:ch:gaps_intro, we review the extreme value statistics of Brownian motion and random walks.

In chapter LABEL:ch:maxk, we consider the order statistics of random walks. We obtain an exact formula for the time to reach the $k^{\rm th}$ maximum of the walk. We review some properties of the distribution of the value taken by the $k^{\rm th}$ maximum of a random walk with finite variance jump distribution. We introduce and compute the quenched and annealed density of maxima both for finite variance and Lévy flights. These results are still unpublished.

In chapter LABEL:ch:gapk, we consider the gap statistics of random walks. We show how to obtain exactly the probability distribution function of the gaps for the random walk with Laplace distribution of jumps. We argue from numerical simulations that this result is universal in the large $n$ limit for any jump distribution with finite variance.

We recently submitted for peer-review an article on the gap statistics of random walks:

LABEL:Art:gap Gap statistics close to the quantile of a random walk,

B. Lacroix-A-Chez-Toine, S. N. Majumdar, G. Schehr,

arXiv preprint, arXiv: 1812.08543, (2018).

Note finally that we recently submitted another article for peer-review, which is largely disconnected to the extreme value statistics:

LABEL:Art:BMcoi *Distribution of Brownian coincidences

*A. Krajenbrink, B. Lacroix-A-Chez-Toine, P. Le Doussal,

arXiv preprint, arXiv: 1903.06511, (2019).

In this article, we compute the distribution of coincidence time ${\cal T}_{N}(t)$, i.e. the total local time of all pairwise coincidences, of $N$ independent Brownian walkers. We show that this problem is related to (i) the Lieb-Liniger problem of $N$ hard-core interacting bosons [PhysRev.130.1605] and (ii) the moments of the canonical partition function of directed polymers (in direct relation to the Kardar-Parisi-Zhang equation [kardar1986dynamic]). We obtain the exact distribution of ${\cal T}_{N}(t)$ for $N=2,3$ for several initial and final conditions and the asymptotic behaviours of the distribution for any values of $N$.