Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes

Gaultier Lambert; Kurt Johansson

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Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes

Kurt Johansson, Gaultier Lambert

Probability · Submitted 24 Apr 2015

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The authors' main claim On the other hand, in the critical regime where the scales are the same, we get a non-Gaussian process in the limit.

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arXiv ID1504.06455 (version 2)

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CitationGaultier Lambert, Kurt Johansson. "Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes." arXiv:1504.06455 (2015). https://pith.science/paper/OFFZIMK7

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We study mesoscopic linear statistics for a class of determinantal point processes which interpolates between Poisson and Gaussian Unitary Ensemble statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the GUE eigenvalue process. An example of such a system comes from considering the distribution of non-colliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the GUE kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we get a central limit theorem of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we get a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study mesoscopic linear statistics for a class of determinantal point processes which interpolates between Poisson and Gaussian Unitary Ensemble statistics.	contribution 0.99	0.88	0.08	asserted 0.99
s1	These processes are obtained by modifying the spectrum of the correlation kernel of the GUE eigenvalue process.	method 0.73	0.53	0.03	asserted 1.00
s2	An example of such a system comes from considering the distribution of non-colliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature.	background 0.48	0.20	0.03	asserted 0.90
s3	When the scale of the modification of the spectrum of the GUE kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we get a central limit theorem of either Poisson or GUE type.	result 1.00	0.88	0.29	asserted 1.00
s4	On the other hand, in the critical regime where the scales are the same, we get a non-Gaussian process in the limit.	result 1.00	0.89	0.34	asserted 1.00
s5	Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics.	result 0.99	0.86	0.14	asserted 1.00
s6	These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process.	method 1.00	0.65	0.05	asserted 1.00
s7	Analogous determinantal processes on the circle are also considered with similar results.	contribution 0.67	0.82	0.03	asserted 1.00

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Original abstract

We study mesoscopic linear statistics for a class of determinantal point processes which interpolates between Poisson and Gaussian Unitary Ensemble statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the GUE eigenvalue process. An example of such a system comes from considering the distribution of non-colliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the GUE kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we get a central limit theorem of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we get a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.

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The proof relies on lemma [ref] and it is straightforward to generalize the argument of [ref] to obtain ([ref]).

Mesoscopic fluctuations for unitary invariant ensembles · Proof of theorem [ref] · 0.94

Uses its method or tool 3 papers

The proof relies on ideas developed in [ref] .

Mesoscopic fluctuations for unitary invariant ensembles · Introduction and results · 0.89
By lemma [ref], $^n_N[F] = ϖ^n_N(F(^(N)))$ for any $n≥2$ and by ([ref]), we obtain [equation] Observe that by lemma [ref], if $| Ł_ij | ≤C exp(αmax{|i|, |j|})$, then $| ϖ^n_0(F(Ł)) | ≤n! exp( n C_F)$ and by a standard argument, see e.g. Lemma 4.8 in [ref] , there exists a random variable whose cumulants are given by $ϖ^n_0(F(Ł))$ for all $n≥2$.

CLT for biorthogonal ensembles and related combinatorial identities · Gaussian fluctuations: proof of theorem [ref] and theorem [ref]. · 0.90
The argument is the same as the proof of Lemma 1 in Soshnikov's paper [ref] on linear statistics of the CUE, see also [ref] or section 2.1 in [ref] .

Incomplete determinantal processes: from random matrix to Poisson statistics · Transition for the 1-dimensional log-gases - the mesoscopic regime. · 0.96

Compares with it 1 paper

However, to the author's knowledge, in contrast to the incomplete ensembles considered in this note, the critical processes discovered in [ref] cannot be described in simple terms like [ref].

Incomplete determinantal processes: from random matrix to Poisson statistics · introduction · 0.98

Cites it as background 6 papers

The cumulants method to analyze the asymptotic distribution of linear statistic of determinantal processes goes back to the work of Costin and Lebowitz, [ref] , for count statistics of the sine process and the general theory was developed by Soshnikov, [ref] , and subsequently applied to many different ensembles coming from random matrix theory, see for instance [ref] .

Incomplete determinantal processes: from random matrix to Poisson statistics · Outline of the proof · 0.82
See e.g. [ref] .

Free fermions and the classical compact groups · Introduction · 0.88
See for instance [ref] for a non-exhaustive list.

Monte Carlo with Determinantal Point Processes · A central limit theorem for multivariate OP Ensembles · 0.87

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This situation is similar to Dyson's Brownian motion which was investigated in [ref] .

On mesoscopic equilibrium for linear statistics in Dyson's Brownian Motion · Discussion and Examples · 0.89

Cites as background 4 papers

This approach has been applied to many classical models in 1 or 2 dimensions using a wide range of techniques, see e.g. [ref] .

Central Limit Theorems for Biorthogonal Ensembles and Asymptotics of Recurrence Coefficients · Discussion and Examples · 0.82
This approach has been applied to many classical models in 1 or 2 dimensions using a wide range of techniques, see e.g. [ref] .

Universality of mesoscopic fluctuations for orthogonal polynomial ensembles · Discussion and Examples · 0.82
This process behaves roughly like Dyson Brownian motion [ref] and this provides a heuristic description of the transition.

On mesoscopic equilibrium for linear statistics in Dyson's Brownian Motion · The Modified Gaussian Unitary Ensembles · 0.80

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