Extreme statistics and index distribution in the classical 1<i>d</i>Coulomb gas

Dhar, Abhishek; Kundu, Anupam; Majumdar, Satya N.; Sabhapandit, Sanjib; Schehr, Grégory

doi:10.1088/1751-8121/aac75f

Cited by 42 publications

(78 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These methods however often appear disconnected: in this Letter we unveil relations between some of them, valid at the edge. Strongly perturbed Coulomb gas are interesting correlated systems by themselves, extensively studied recently [52], but not much at their edge [53].…”

mentioning

confidence: 99%

Linear statistics and pushed Coulomb gas at the edge of β -random matrices: Four paths to large deviations

Krajenbrink¹,

Doussal²

2019

EPL

View full text Add to dashboard Cite

The Airy β point process, ai ≡ N 2/3 (λi − 2), describes the eigenvalues λi at the edge of the Gaussian β ensembles of random matrices for large matrix size N → ∞. We study the probability distribution function (PDF) of linear statistics L = i tϕ(t −2/3 ai) for large parameter t. We show the large deviation forms E Airy,β [exp(−L)] ∼ exp(−t 2 Σ[ϕ]) and P (L) ∼ exp(−t 2 G(L/t 2 )) for the cumulant generating function and the PDF. We obtain the exact rate function Σ[ϕ] using four apparently different methods (i) the electrostatics of a Coulomb gas (ii) a random Schrödinger problem, i.e. the stochastic Airy operator (iii) a cumulant expansion (iv) a non-local non-linear differential Painlevé type equation. Each method was independently introduced previously to obtain the lower tail of the Kardar-Parisi-Zhang equation [1][2][3][4]. Here we show their equivalence in a more general framework. Our results are obtained for a class of functions ϕ, the monotonous soft walls, containing the monomials ϕ(x) = (u + x) γ + and the exponential ϕ(x) = e u+x and equivalently describe the response of a Coulomb gas pushed at its edge. The small u behavior of the excess energy Σ[ϕ] exhibits a change at γ = 3/2 between a non-perturbative hard wall like regime for γ < 3/2 (third order free-to-pushed transition) and a perturbative deformation of the edge for γ > 3/2 (higher order transition). Applications are given, among them: (i) truncated linear statistics such as N 1 i=1 ai, leading to a formula for the PDF of the ground state energy of N1 1 noninteracting fermions in a linear plus random potential (ii) ∼ (β − 2)/r 2 interacting spinless fermions in a trap at the edge of a Fermi gas (iii) traces of large powers of random matrices.√ β 2 v(x), H SAO = t −2/3 H SAO with energy levels b i = t −2/3 ε i = −t −2/3 a i and obtain

show abstract

mentioning

confidence: 99%

Linear statistics and pushed Coulomb gas at the edge of β -random matrices: Four paths to large deviations

Krajenbrink¹,

Doussal²

2019

EPL

View full text Add to dashboard Cite

show abstract

“…for which the equilibrium measure is the flat distribution and the distribution of its largest (lowest) eigenvalue have recently been computed exactly, both for typical fluctuations and also for large deviations [58,59,60].…”

Section: Relations To Other Modelsmentioning

confidence: 99%

Stability of large complex systems with heterogeneous relaxation dynamics

Mergny,

Majumdar

2021

Preprint

View full text Add to dashboard Cite

We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size ni (with respect to its equilibrium value):The ai > 0's are the intrinsic decay rates, Jij is a real symmetric (N × N ) Gaussian random matrix and √ T measures the strength of pairwise interaction between different species. Unlike in May's original homogeneous model, each species has now an intrinsic damping ai that may differ from one another. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = Tc. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian Motion (DBM), starting at the initial position ai and evolving in 'time' T . In the large N → ∞ limit, using this DBM picture, we are able to completely characterize Tc for arbitrary density µ(a) of the ai's. For a specific flat configuration ai = 1 + σ i−1 N , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N , we also compute the large deviation properties of the probability of stability on the stable side T < Tc using a Coulomb gas representation.

show abstract

“…The harmonic term tries to push the charges close to the origin, while the repulsive interaction tries to spread them apart. The competition between these two terms leads to interesting collective properties of these charges [4,5,6,7,8,9]. Given this energy function in Eq.…”

Section: Introductionmentioning

confidence: 99%

“…where the left and the right large deviation functions Φ − (w) and Φ + (w) were also computed explicitly [7,8]. The left rate function is given by [7,8] Φ…”

Section: Introductionmentioning

confidence: 99%

Truncated linear statistics in the one dimensional one-component plasma

Flack,

Majumdar,

Schehr

2021

Preprint

View full text Add to dashboard Cite

In this paper, we study the probability distribution of the observable s = (1/N )representing the ordered positions of N particles in a 1d one-component plasma, i.e., N harmonically confined charges on a line, with pairwise repulsive 1d Coulomb interaction |x i − x j |. This observable represents an example of a truncated linear statistics -here the center of mass of the N = κ N (with 0 < κ ≤ 1), rightmost particles. It interpolates between the position of the rightmost particle (in the limit κ → 0) and the full center of mass (in the limit κ → 1). We show that, for large N , s fluctuates around its mean s and the typical fluctuations are Gaussian, of width O(N −3/2 ). The atypical large fluctuations of s, for fixed κ, are instead described by a large deviation form P N,κ (s) exp −N 3 φ κ (s) , where the rate function φ κ (s) is computed analytically. We show that φ κ (s) takes different functional forms in five distinct regions in the (κ, s) plane separated by phase boundaries, thus leading to a rich phase diagram in the (κ, s) plane. Across all the phase boundaries the rate function φ(κ, s) undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.

show abstract

Extreme statistics and index distribution in the classical 1dCoulomb gas

Cited by 42 publications

References 57 publications

Linear statistics and pushed Coulomb gas at the edge of β -random matrices: Four paths to large deviations

Linear statistics and pushed Coulomb gas at the edge of β -random matrices: Four paths to large deviations

Stability of large complex systems with heterogeneous relaxation dynamics

Truncated linear statistics in the one dimensional one-component plasma

Contact Info

Product

Resources

About