Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential

Rojas, Rafael Díaz Hernández; Calva, Christopher Sebastian Hidalgo; Castillo, Isaac Pérez

doi:10.1103/physreve.98.020104

Cited by 10 publications

(8 citation statements)

References 40 publications

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“…which can be checked by changing the variables k → − k in the definition of E α (x, n) in equation (54). In terms of these two functions, one can easily see that equation ( 52) reads…”

Section: Typical Fluctuations Of the Gapmentioning

confidence: 99%

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Gap probability and full counting statistics in the one-dimensional one-component plasma

Flack¹,

Majumdar²,

Schehr³

2022

J. Stat. Mech.

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We consider the 1d one-component plasma in thermal equilibrium, consisting of N equally charged particles on a line, with pairwise Coulomb repulsion and confined by an external harmonic potential. We study two observables: (i) the distribution of the gap between two consecutive particles in the bulk and (ii) the distribution of the number of particles N I in a fixed interval I = [−L, +L] inside the bulk, the so-called full-counting-statistics (FCS). For both observables, we compute, for large N, the distribution of the typical as well as atypical large fluctuations. We show that the distribution of the typical fluctuations of the gap g is described by the scaling form P gap,bulk ( g , N ) ∼ N H α ( g N ) , where α is the interaction coupling and the scaling function H α (z) is computed explicitly. It has a faster than Gaussian tail for large z: H α ( z ) ∼ e − z 3 / ( 96 α ) as z → ∞. Similarly, for the FCS, we show that the distribution of the typical fluctuations of N I is described by the scaling form P FCS ( N I , N ) ∼ 2 α U α [ 2 α ( N I − N ¯ I ) ] , where N ¯ I = L N / ( 2 α ) is the average value of N I and the scaling function U α (z) is obtained explicitly. For both observables, we show that the probability of large fluctuations is described by large deviations forms with respective rate functions that we compute explicitly. Our numerical Monte-Carlo simulations are in good agreement with our analytical predictions.

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“…which can be checked by changing the variables k → − k in the definition of E α (x, n) in equation (54). In terms of these two functions, one can easily see that equation ( 52) reads…”

Section: Typical Fluctuations Of the Gapmentioning

confidence: 99%

“…The denominator D α (∞, M), by definition in equation (54), is simply proportional to the partition function Z M of the 1dOCP gas with M particles. More precisely, it is easy to see that…”

Section: Typical Fluctuations Of the Gapmentioning

confidence: 99%

Gap probability and full counting statistics in the one-dimensional one-component plasma

Flack¹,

Majumdar²,

Schehr³

2022

J. Stat. Mech.

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“…Importantly, they noticed that the deviations to the left of, say, the largest eigenvalues are markedly different to the ones on its right, scaling differently with the system size. More research was done along these lines for other standard random matrix ensembles [20][21][22][23][24][25][26][27][28][29], in diluted ensembles of random matrices [30][31][32][33][34][35], and generalizations based on Dyson's log-gas analogy [36][37][38], among many others, to ascertain the robustness of this new emergent law on the statistics of extreme values.…”

Section: Introductionmentioning

confidence: 99%

A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles

Carro¹,

Benet²,

Castillo³

2022

Preprint

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In spite of its simplicity, the central limit theorem captures one of the most outstanding phenomena in mathematical physics, that of universality. It states that, while navigating the set of all possible distributions by their convolution there exists a fixed point, the family of Gaussian distributions, with a fairly large basin of attraction comprising a wide family of weakly correlated distributions. More colloquially, this means roughly speaking that the sum of independent and identically distributed random variables follows a Gaussian distribution irrespective of the details of the original distributions from which these random variables are drawn. While this classical result is well understood, and is the reason behind many physical phenomena, it is still not very clear what happens to this universal behaviour when the random variables become correlated. Not a general overarching theorem exists on a possible emergence of new universal behaviour and, so far as we are aware of, this area of research must be explored in a case-by-case basis. A fruitful mathematically laboratory to investigate the rising of new universal properties is offered by the set of eigenvalues of random matrices. In this regard a lot of work has been done using the standard random matrix ensembles and focusing on the distribution of extreme eigenvalues. In this case, the distribution of the largest -or smallest-eigenvalue departs from the Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom distribution. One may wonder, yet again, how robust is this new universal behaviour captured by the Tracy-Widom distribution when the correlation among eigenvalues changes. Few answers have been provided to this poignant question and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we study numerically the probability distribution for the normalized largest eigenvalue of the interacting k-body fermionic orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We find a smooth transition from a slightly asymmetric Gaussian-like distribution, for small k/m, to the Tracy-Widom distribution as k/m → 1, where k is the rank of the interaction and m is the number of fermions. Correlations at the edge of the spectrum are stronger for small values of k/m, and are independent of the number of particles considered. Our results indicate that subtle correlations towards the edge of the spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the dilute limit, from those expected for the standard random matrix ensembles.

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“…with the support x ∈ [−N J, N J]. The statistics of the position of the rightmost particle x max has been studied recently [33][34][35][36]. Its average is x max = N J and its typical fluctuations are O(1) and are governed by the CDF F (−1) β (x) which is a solution to a non-local eigenvalue equation…”

Section: Introductionmentioning

confidence: 99%

Edge fluctuations and third-order phase transition in harmonically confined long-range systems

Kethepalli,

Kulkarni,

Kundu

et al. 2021

Preprint

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We study the distribution of the position of the rightmost particle x max in a N -particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form r −k with −2 < k < ∞ where r is the inter-particle distance. In equilibrium at temperature O(1), the gas settles on a finite length scale L N that depends on N and k. We numerically observe that the typical fluctuation of y max = x max /L N around its mean is of O(N −η k ). Over this length scale, the distribution of the typical fluctuations has a N independent scaling form. We show that the exponent η k obtained from the Hessian theory predicts the scale of typical fluctuations remarkably well. The distribution of atypical fluctuations to the left and right of the mean y max are governed by the left and right large deviation functions, respectively. We compute these large deviation functions explicitly ∀k > −2. We also find that these large deviation functions describe a pulled to pushed type phase transition as observed in Dyson's log-gas (k → 0) and 1d one component plasma (k = −1). Remarkably, we find that the phase transition remains 3 rd order for the entire regime. Our results demonstrate the striking universality of the 3 rd order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the large deviation functions via Monte Carlo simulation using an importance sampling algorithm.

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Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential

Cited by 10 publications

References 40 publications

Gap probability and full counting statistics in the one-dimensional one-component plasma

Gap probability and full counting statistics in the one-dimensional one-component plasma

A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles

Edge fluctuations and third-order phase transition in harmonically confined long-range systems

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