Statistics of fermions in a d-dimensional box near a hard wall

Lacroix-A-Chez-Toine, Bertrand; Doussal, Pierre Le; Majumdar, Satya N.; Schehr, Grégory

doi:10.1209/0295-5075/120/10006

Cited by 30 publications

(82 citation statements)

References 35 publications

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“…Pursuing this analogy with random matrices, one may wonder whether one can define a "density" associated to this point process that would capture the existence of these edge regions, and would be the equivalent of the Wigner semi-circle in random matrix theory. This is left for future investigations [46]. which is clearly different from the scaling function found at the edge [20] (corresponding to the limit α → 0), given in Eq.…”

Section: Resultsmentioning

confidence: 58%

“…What happens for heavy tailed jump distributions, i.e. the case of Lévy flights, remains a challenging open question, in particular because we do not know how to solve the backward integral equations (45) and (46) in this case. We hope that the results obtained here will motivate further works to develop alternative methods to study the gap statistics of Lévy flights.…”

Section: Resultsmentioning

confidence: 99%

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Gap statistics close to the quantile of a random walk

Lacroix-A-Chez-Toine

Majumdar

Schehr

2019

J. Phys. A: Math. Theor.

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We consider a random walk of n steps starting at x 0 = 0 with a double exponential (Laplace) jump distribution. We compute exactly the distribution p k,n (∆) of the gap d k,n between the k th and (k + 1) th maxima in the limit of large n and large k, with α = k/n fixed. We show that the typical fluctuations of the gaps, which are of order O(n −1/2 ), are described by a universal α-dependent distribution, which we compute explicitly.Interestingly, this distribution has an inverse cubic tail, which implies a non-trivial n-dependence of the moments of the gaps. We also argue, based on numerical simulations, that this distribution is universal, i.e. it holds for more general jump distributions (not only the Laplace distribution), which are continuous, symmetric with a well defined second moment. Finally, we also compute the large deviation form of the gap distribution pαn,n(∆) for ∆ = O(1), which turns out to be non-universal. arXiv:1812.08543v1 [cond-mat.stat-mech]

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Section: Resultsmentioning

confidence: 58%

Section: Resultsmentioning

confidence: 99%

Gap statistics close to the quantile of a random walk

Lacroix-A-Chez-Toine

Majumdar

Schehr

2019

J. Phys. A: Math. Theor.

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“…Another interesting feature of the case d > 1 concerns the large deviations of r max (T ). Indeed, it was shown in [235,236] that, at variance with the standard scenario (94) found for instance for the β-Gaussian ensembles, there exists generically an intermediate regime of fluctuations between the typical and the left large deviation regime. This intermediate regime, which is actually not restricted to EVS but also concerns other observables like the full counting statistics [237], seems to be a rather generic feature of higher dimensional determinantal processes, including in particular (complex) Ginibre random matrices and two-dimensional Coulomb gas [238].…”

Section: Extreme Statistics Of Trapped Fermionsmentioning

confidence: 84%

“…The statistics of the maximal radial distance r max (T ) of the fermions from the trap center was also studied in dimension d > 1, both for smooth potentials like the harmonic well [232], and the hard box (spherical) potential [235,236]. In both cases, although the positions of the fermions are strongly correlated at T = 0, it was found that, for large N , the statistics of r max (T = 0) is given by the EVS of IID random variables, i.e.…”

Section: Extreme Statistics Of Trapped Fermionsmentioning

confidence: 99%

Extreme value statistics of correlated random variables: A pedagogical review

2020

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Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of 'uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this review, we will first recall the classical EVS for uncorrelated variables and discuss the three universality classes of extreme value limiting distribution, known as the Gumbel, Fréchet and Weibull distribution. We then show that, for weakly correlated random variables with a finite correlation length/time, the limiting extreme value distribution can still be inferred from that of the uncorrelated variables using a renormalisation group-like argument. Finally, we consider the most interesting examples of strongly correlated variables for which there are very few exact results for the EVS. We discuss few examples of such strongly correlated systems (such as the Brownian motion and the eigenvalues of a random matrix) where some analytical progress can be made. We also discuss other observables related to extremes, such as the density of near-extreme events, time at which an extreme value occurs, order and record statistics, etc.

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“…It is natural to ask the question if there exist quantum potentials that correspond to the general JUE with arbitrary parameters a and b. Indeed, it was shown recently [38] that a potential of the type…”

Section: Hard Box Potential and The Juementioning

confidence: 99%

Noninteracting fermions in a trap and random matrix theory

Dean¹,

Doussal²,

Majumdar³

et al. 2019

J. Phys. A: Math. Theor.

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112

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We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions N 1, the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i. e. close to the center of the trap, are well described by standard manybody techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-toone correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the "Airy kernel") and hard edge (described by the "Bessel kernel") universality classes. We further discuss extensions of these results to higher dimensions d ≥ 2 and to finite temperature. Finally, we discuss correlations in the phase space, i.e., in the space of positions and momenta, characterized by the so called Wigner function. arXiv:1810.12583v1 [cond-mat.stat-mech]

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Statistics of fermions in a d-dimensional box near a hard wall

Cited by 30 publications

References 35 publications

Gap statistics close to the quantile of a random walk

Gap statistics close to the quantile of a random walk

Extreme value statistics of correlated random variables: A pedagogical review

Noninteracting fermions in a trap and random matrix theory

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