On the Critical–Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective

Keating, Jon P; Wong, Mo Dick

doi:10.1007/s00220-022-04429-3

Cited by 3 publications

(2 citation statements)

References 86 publications

(234 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The set A k,β;l and the function f (v; l) are defined in the proof, see (2. 19) and (2.26) respectively. Also, the µ n are defined in terms of the θ m in (2.16).…”

Section: The Symplectic Group Sp(2n )mentioning

confidence: 97%

“…Lastly, we mention the recent work of Keating and Wong [19] who, through the perspective of Gaussian multiplicative chaos, have obtained an asymptotic formula for MoM U (N ) (k, β) at the critical point kβ 2 = 1 for k ≥ 2 an integer. Their result confirms that the moments of moments are of the order N log N as N → ∞ and they conjecture that this asymptotic result holds for all k > 1 and provide a heuristic argument in support of this.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Random matrix theory and moments of moments of L-functions

Andrade

Best

2022

Random Matrices: Theory Appl.

View full text Add to dashboard Cite

In this paper, we give an analytic proof of the asymptotic behavior of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments of [Formula: see text]-functions with symplectic and orthogonal symmetry. Specifically, we show that these conjectures follow from the shifted moments conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith.

show abstract

“…The set A k,β;l and the function f (v; l) are defined in the proof, see (2. 19) and (2.26) respectively. Also, the µ n are defined in terms of the θ m in (2.16).…”

Section: The Symplectic Group Sp(2n )mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Random matrix theory and moments of moments of L-functions

Andrade

Best

2022

Random Matrices: Theory Appl.

View full text Add to dashboard Cite

show abstract

Freezing transition and moments of moments of the riemann zeta function

Curran

2024

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

Moments of moments of the Riemann zeta function, defined by $$ \text{MoM}_T(k,\beta) := \frac{1}{T}\int_T^{2T} \Bigg(\,\int\limits_{ |h|\leq (\log T)^\theta}|\zeta(\frac{1}{2} + i t + ih)|^{2\beta}\ dh\Bigg)^k\ dt, $$ where $k,\beta \geq 0$ and $\theta \gt -1$ were introduced by Fyodorov and Keating, Freezing transitions and extreme values: random matrix theory, and disordered landscapes, Philos. Trans. Roy. Soc. A: 372 no. 2007 (2014), 20120503 A doi:10.1098/rsta.2012.0503 when comparing extreme values of zeta in short intervals to those of characteristic polynomials of random unitary matrices. We study the k = 2 case as $T \rightarrow \infty$ and obtain sharp upper bounds for $\text{MoM}_T(2,\beta)$ for all real $0\leq \beta \leq 1$ as well as lower bounds of the conjectured order for all $\beta \geq 0$. In particular, we show that the second moment of moments undergoes a freezing phase transition with critical exponent $\beta = \frac{1}{\sqrt{2}}$. The main technical ingredient is a new estimate for the correlation of two shifted $2\beta^{\text{th}}$ powers of zeta with $0\leq \beta \leq 1$.

show abstract

On the moments of moments of random matrices and Ehrhart polynomials

Assiotis¹,

Eriksson²,

Ni³

2021

Preprint

View full text Add to dashboard Cite

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.

show abstract

On the Critical–Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective

Cited by 3 publications

References 86 publications

Random matrix theory and moments of moments of L-functions

Random matrix theory and moments of moments of L-functions

Freezing transition and moments of moments of the riemann zeta function

On the moments of moments of random matrices and Ehrhart polynomials

Contact Info

Product

Resources

About