Maximum of the Characteristic Polynomial for a Random Permutation Matrix

Cook, Nicholas A.; Zeitouni, Ofer

doi:10.1002/cpa.21899

Cited by 8 publications

(4 citation statements)

References 40 publications

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“…For the (integer-valued) Discrete Gaussian model, it has been shown that the maximum is of logarithmic order [50]. There are also results for the maximum of the logarithm of the characteristic polynomial of certain random matrices, see for example [4,21,22,46]. Like these random matrix ensembles, the sine-Gordon model is closely related to a Coulomb gas type particle ensemble, but our point of view is completely different.…”

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confidence: 92%

Maximum and coupling of the sine-Gordon field

Bauerschmidt¹,

Hofstetter²

2020

Preprint

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For 0 < β < 6π, we prove that the distribution of the centred maximum of the ε-regularised continuum sine-Gordon field on the two-dimensional torus converges to a randomly shifted Gumbel distribution as ε → 0. Our proof relies on a strong coupling at all scales of the sine-Gordon field with the Gaussian free field, of independent interest, and extensions of existing methods for the maximum of the lattice Gaussian free field.

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confidence: 92%

Maximum and coupling of the sine-Gordon field

Bauerschmidt¹,

Hofstetter²

2020

Preprint

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“…For the characteristic polynomial of the GUE, as well as other Hermitian unitary invariant ensembles, the law of large numbers for the maximum of the absolute value of the characteristic polynomial was obtained in [33]. Cook and Zeitouni [16] also obtained a law of large numbers for the maximum of the characteristic polynomial for a random permutation matrix, in which case their result does not match with the prediction from Gaussian log-correlated field because of arithmetic effects. Finally, in the article [15] in preparation with Claeys, Fahs and Webb, we obtain the counterpart of Theorem 1.1 for the imaginary part of the characteristic polynomial of a large class of Hermitian unitary invariant ensembles.…”

Section: Comments On Theorem and Further Resultsmentioning

confidence: 99%

Maximum of the characteristic polynomial of the Ginibre ensemble

Lambert

2019

Preprint

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We compute the leading asymptotics of the maximum of the (centered) logarithm of the absolute value of the characteristic polynomial, denoted Ψ , of the Ginibre ensemble as the dimension of the random matrix tends to +∞. The method relies on the log-correlated structure of the field Ψ and we obtain the lower-bound for the maximum by constructing a family of Gaussian multiplicative chaos measures associated with certain regularization of Ψ at small mesoscopic scales. We also obtain the leading asymptotics for the dimensions of the sets of thick points and verify that they are consistent with the predictions coming from the Gaussian Free Field. A key technical input is the approach from [3] to derive the necessary asymptotics,

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“…This is obtained by taking t = 1, s = λ + 1 and A = V N \V δ N in (25). The following lemma, which will be proved in Section 3, provides the upper bound for the left tail of the maximum restricted to a sub-box of V N .…”

Section: P(maxmentioning

confidence: 99%

“…For the counterpart progress on the Riemann zeta side, we refer the reader to [7,8,11]. For random permutation matrices, a law of large numbers result was obtained in [25].…”

Section: Introductionmentioning

confidence: 99%

Universality of Poisson-Dirichlet law for log-correlated Gaussian fields via level set statistics

Ganguly¹,

Nam²

2023

Preprint

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Many low temperature disordered systems are expected to exhibit Poisson-Dirichlet (PD) statistics. In this paper, we focus on the case when the underlying disorder is a logarithmically correlated Gaussian process φN on the box [−N, N ] d ⊂ Z d . Canonical examples include branching random walk, * -scale invariant fields, with the central example being the two dimensional Gaussian free field (GFF), a universal scaling limit of a wide range of statistical mechanics models.The corresponding Gibbs measure obtained by exponentiating β (inverse temperature) times φN is a discrete version of the Gaussian multiplicative chaos (GMC) famously constructed by Kahane [37]. In the low temperature or supercritical regime, i.e., β larger than a critical βc, the GMC is expected to exhibit atomic behavior on suitable renormalization, dictated by the extremal statistics or near maximum values of φN . Moreover, it is predicted going back to a conjecture made in 2001 in [23], that the weights of this atomic GMC has a PD distribution.In a series of works culminating in [19], Biskup and Louidor carried out a comprehensive study of the near maxima of the 2D GFF, and established the conjectured PD behavior throughout the super-critical regime (β > 2). In another direction, in [28], Ding, Roy and Zeitouni established universal behavior of the maximum for a general class of log-correlated Gaussian fields.In this paper we continue this program simply under the assumption of log-correlation and nothing further. We prove that the GMC concentrates on an O(1) neighborhood of the local extrema and the PD prediction made in [23] holds, in any dimension d, throughout the supercritical regime β > √ 2d, significantly generalizing past results. While many of the arguments for the GFF make use of the powerful Gibbs-Markov property, in absence of any Markovian structure for general Gaussian fields, we develop and use as our key input a sharp estimate of the size of level sets, a result we believe could have other applications.

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Maximum of the Characteristic Polynomial for a Random Permutation Matrix

Cited by 8 publications

References 40 publications

Maximum and coupling of the sine-Gordon field

Maximum and coupling of the sine-Gordon field

Maximum of the characteristic polynomial of the Ginibre ensemble

Universality of Poisson-Dirichlet law for log-correlated Gaussian fields via level set statistics

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