We review the authors’ recent work where we obtain the uniform large *s* asymptotics for the Fredholm determinant $D(s,\gamma )\u2254det(I\u2212\gamma Ks\u21beL2(\u22121,1))$, 0 ≤ *γ* ≤ 1. The operator *K*_{s} acts with kernel *K*_{s}(*x*, *y*) = sin(*s*(*x* − *y*))/(*π*(*x* − *y*)), and *D*(*s*, *γ*) appears for instance in Dyson’s model of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned Gaussian unitary ensemble.

Dedicated to the memory of Ludwig Faddeev

## I. INTRODUCTION

Consider the determinantal point process {*x*_{j}} on the real line with a correlation kernel

It is well known that this process appears in the bulk scaling limit for random matrices (in particular, in the Gaussian Unitary Ensemble, GUE) where the average distance between particles *x*_{j} is unity. See Ref. 19 and the references therein to Lenard’s work, that for any $\varphi \u2208L\u221e(R)$ whose support is inside a bounded measurable set *B*,

where the determinant on the right-hand side of (1.1) is the Fredholm determinant of the trace-class integral operator *ϕK* on *L*^{2} (*B*) with kernel *ϕ*(*x*)*K*(*x*, *y*). Provided we define

then applying (1.1) to *ϕ*(*x*) = 1 − e^{−W(x)}, we find

where

We can interpret $v$ > 0 above as an external potential applied to the Coulomb log-gas particle system {*x*_{j}} whose effect is to push particles out of the interval $(\u2212s\pi ,s\pi )$. We are interested in *D*(*s*, *γ*) for large *s* and 0 ≤ $v$ ≤ +*∞*. This has been a long standing problem, and we shall first review relevant classical results.

## II. ASYMPTOTIC RESULTS

### A. Fixed and slowly growing $v$

If $v$ = +*∞*, i.e., *γ* = 1, then *D*(*s*, 1) is the probability that there are no sine process particles in the interval $(\u2212s\pi ,s\pi )$. In this case,

where $lnc0=112ln\u20612+3\zeta \u2032(\u22121)$ and *ζ*′(*z*) is the derivative of Riemann’s zeta function. The main term, $e\u221212s2$, in (2.1) was first conjectured by Dyson, followed by a conjecture for $e\u221212s2s\u221214$ by des Cloizeaux and Mehta in Ref. 12. A full asymptotic expansion for *D*(*s*, 1), including the numerical value of *c*_{0}, was then identified by Dyson^{15} in 1976 who used inverse scattering techniques for Schrödinger operators and an earlier study of Widom^{21} on Toeplitz determinants. Dyson’s calculations were not fully rigorous, and the first proof of the main term in (2.1) was given by Widom^{22} in 1994. A proof of the full expansion (except for the value of *c*_{0}) was carried out in Ref. 14, and *c*_{0}, the so-called Widom-Dyson constant, was finally proved in three different ways in Refs. 13, 17, and 18.

If $0\u2264v<s13$ (which includes the case of fixed finite *v*), there exist constants *s*_{0}, *c*_{j} > 0 such that for *s* > *s*_{0},

where *G*(*z*) is the Barnes G-function

with Euler’s constant *γ*_{E}, and

Expansion (2.2) in the case of fixed finite $v$ as *s* → +*∞* and with *r*(*s*, $v$) replaced by $Os\u22121$ was first established by Basor and Widom in 1983^{1} and later, independently, by Budylin and Buslaev.^{10} It was extended to the presented range with varying $v$ in Ref. 7. At this point, it is worthwhile to contrast the Gaussian decay in the leading order of (2.1) with the leading exponential decay (for fixed $v$) in (2.2). The underlying non-trivial transition (as worked out in the authors’ studies, Refs. 6–8) is summarized in Subsection II B.

### B. Faster growing $v$

We shall now describe the asymptotic transition between (2.1) and (2.2) which takes place when $v$ is allowed to grow faster with *s* than $s13$. In more detail, we describe the large *s* asymptotics of *D*(*s*, *γ*) in the quarter-plane (*s*, $v$), *s*, $v$ > 0; see Fig. 1. Heuristically, this problem was first addressed by Dyson in Ref. 16 who discovered an oscillatory behavior in the asymptotics and the appearance of Jacobi-theta functions (see Ref. 6 for a discussion of Dyson’s calculations). Let

so that *γ* = 1 − $e\u22122v$ = 1 − e^{−2κs}. First, we have

*As s*→

*∞ with*$\kappa >1\u221214ln\u2061ss$,

*where the expansion for*D(

*s*, 1)

*is given in*(2.1).

*Moreover, when s*→ ∞

*with*

*where*[

*x*]

*denotes the integer part of*$x\u2208R$,

Observe that $\kappa =1\u2212u2ln\u2061ss$ can be written as $v=s\u2212u2ln\u2061s$ which implies that $e2(s\u2212v)$ = *s*^{u}. Thus, if $q\u221212\u2264u<q+12$, only the factors indexed with *j* = 0, 1, …, *q* − 1 contribute to the product in (2.4), while terms with larger indices *j* yield a decaying with *s* contribution. In geometric terms, if one moves to the right in Fig. 1, each time a *Stokes curve* $v=s\u221214(2k+1)ln\u2061s$ is crossed, a factor is added to (2.4). One can roughly interpret this phenomenon as a particle jumping to the center of the interval (−1, 1), i.e., the rescaled original interval $(\u2212s\pi ,s\pi )$. Note that for $v>s\u221214ln\u2061s$, estimate (2.3) shows that the leading order asymptotics of *D*(*s*, *γ*) are the same as for the case $v$ = *∞*. The asymptotic expansions (2.3) and (2.4) in the case of fixed *q* (rather than growing with *s*, as stated in Theorem 2.1) were proven in Ref. 6 using general operator theoretical arguments and a result of Slepian^{20} on the asymptotics of the eigenvalues of the sine-kernel integral operator. Each factor in the product (2.4) is related to an eigenvalue of the associated integral operator. In full generality, Theorem 2.1 is proven in Ref. 8 using Riemann-Hilbert nonlinear steepest descent techniques.

*q*can grow with

*s*as $[(ln\u2061s)13]$ in (2.4) (this growth estimate can be improved) allows us to connect the asymptotics of Theorem 2.1 with the asymptotics in the lower region of the (

*s*, $v$) quarter-plane, as described in the following. Let 0 <

*a*=

*a*(

*κ*) < 1 be the (unique!) solution of the equation

*K*and

*E*,

*a*∈ (0, 1) and complementary modulus $a\u2032\u22541\u2212a2$. We also require the third Jacobi theta-function

*θ*(

*z*+ 1|

*τ*) =

*θ*(

*z*|

*τ*), and moreover has the property

*θ*(

*z*+

*τ*|

*τ*) =

*θ*(

*z*|

*τ*)e

^{−iπτ−2πiz}, i.e., its second logarithmic derivative with respect to

*z*is an elliptic function with periods 1 and

*τ*. We then have

*As s*→

*∞ with*$0<\kappa <1\u221214(ln\u2061s)43s$,

*where*

*and for any δ*> 0,

*there exists C*(

*δ*) > 0

*such that*|

*B*(

*s*, $v$)| ≤

*C*(

*δ*)

*for*0 <

*κ*< 1 −

*δ as s*→

*∞*.

*Moreover,*

*and*

*B*(

*s*, $v$) in (2.5) in terms of Jacobi theta-functions is given in Refs. 6 and 8; here we are satisfied with the limiting behaviors (2.6) and (2.7). In fact, the asymptotics of Theorems 2.1 and 2.2 overlap in the region

*κ*↑ 1 and thus

*a*↓ 0. Even better, they overlap in a larger region since the estimate on

*κ*from above in Theorem 2.2 can be somewhat increased, while the estimate on

*κ*from below in Theorem 2.1 can be somewhat decreased. Note also that the asymptotics of Theorem 2.2 are valid in the case of fixed $v$ and

*s*→

*∞*when

*κ*↓ 0 and

*a*↑ 1. In this case, the

*κ*↓ 0 expansions are

*θ*(

*sV*|

*τ*) = 1 +

*o*(1), and using (2.7), we immediately recover (2.2) from (2.5), though with a worse error estimate. Theorem 2.2 above was proven in Refs. 6 and 7 for the narrower region 0 <

*κ*< 1 −

*δ*with

*δ*fixed. Its proof relies on Riemann-Hilbert nonlinear steepest descent techniques which were then extended to apply to the full range of Theorem 2.2 in Ref. 8.

Theorems 2.1 and 2.2 provide a full analytic description of the transition between (2.1) and (2.2). In terms of the initial log-gas particle system, the following rough picture for the particles’ behaviors follows from our analysis: First, for $v>s\u221214ln\u2061s$, no particles are expected in the interval (−1, 1). As the rate of increase of $v$ with *s* slows down, particles begin jumping one by one to the center of the interval: $q\u2208Z\u22651$ particles at the interval center correspond to the asymptotics (2.4), and their behavior is described by classical Hermite polynomials; see Ref. 8. As *q* increases, a transition takes place to the theta-function behavior of (2.5), at first with *a* close to zero. The particles then tend to fill in the centermost subinterval (−*a*, *a*) ⊂ (−1, 1), where the soft edges ±*a* are characterized by an approximate Airy function behavior; see Ref. 6. As the growth of $v$ with *s* slows down further, the subinterval (−*a*, *a*) increases and eventually *a* approaches 1. At this point, a transition occurs to the limit *s* → *∞* with $v$ fixed.

## III. THINNED SINE PROCESS

In closing, we recall the following additional interpretation of *D*(*s*, *γ*), noticed by Bohigas and Pato:^{3} (see, the discussion in Ref. 6) The Fredholm determinant *D*(*s*, *γ*) equals the probability that the interval $(\u2212s\pi ,s\pi )$ contains no particles of the *thinned process* obtained from {*x*_{j}} by removing each particle *x*_{j} independently with likelihood 1 − *γ*. Indeed, using the well-known Fredholm determinant expression for the probability of *n* sine-kernel process particles *x*_{j} being in a given interval, we obtain via inclusion-exclusion principle

In short, *D*(*s*, *γ*) is simply the gap-probability of $(\u2212s\pi ,s\pi )$ in the thinned sine-process and thus also the central building block in the evaluation of the thinned Gaudin distribution, i.e., the limiting distribution of gaps between consecutive thinned bulk eigenvalues. Following the arguments of Bohigas and Pato, we contract the *s*-scale in order to keep to unity the average distance between the thinned particles; i.e., instead of *D*(*s*, *γ*), we consider

It follows from (2.2) that at leading order, as *γ* = 1 − $e\u22122v$ ↓ 0,

which is the gap probability of $(\u2212s\pi \gamma ,s\pi \gamma )$ in the particle system obtained from a Poisson point process by removing each particle with probability 1 − *γ* [in order to obtain (3.1), Bohigas and Pato used the asymptotics for fixed $v$, but a rigorous analysis requires the result for varying *γ*, hence varying $v$; see Ref. 6]. Indeed, the gap probability in this thinned Poissonian system is exactly given by

On the other hand, for *γ* = 1, we obtain $D(s\gamma ,\gamma )\u223ce\u221212s2$, i.e., by thinning bulk eigenvalues in the GUE, we can interpolate between a Poissonian and a random matrix theory system. For other matrix ensembles, such as the Circular Unitary Ensemble, CUE, corresponding gap probability phenomena have been studied in Ref. 11. For interpolation mechanisms between random matrix theory systems and Weibull modelled particle systems, we refer the interested reader to Refs. 2, 5, 4, and 9.

## ACKNOWLEDGMENTS

T.B. acknowledges support of the AMS and the Simons Foundation through a travel grant. The work of P.D. is supported under NSF Grant No. DMS-1300965, and A.I. acknowledges support of the NSF Grant No. DMS-1700261 and the Russian Science Foundation Grant No. 17-11-01126. I.K.’s work is supported by the Leverhulme Trust research project Grant No. RPG-2018-260.