On the mean density of states of some matrices related to the beta ensembles and an application to the Toda lattice

Mazzuca, Guido

doi:10.1063/5.0076539

Cited by 9 publications

(10 citation statements)

References 31 publications

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“…We notice that, according to (47), all the a j are independent. Hence, in order to obtain the density of states in the low-temperature limit, we have to compute the weak limit of the Proceeding as in the previous cases, it follows that the following limit holds for all bounded and continuous f : D → R:…”

Section: Parameter Limitmentioning

confidence: 98%

“…The Lax matrix L in (16) becomes a random matrix when the entries are sampled according to (17). Such random matrix can be linked to the so-called Laguerre α-ensemble [47]. The connection is obtained noticing that the matrix L admits the following decomposition…”

Section: Laguerre ˇ-Ensemble and The Exponential Toda Latticementioning

confidence: 99%

“…In order to study the density of states of the exponential Toda lattice, we need the following result proved in [47]. (21).…”

Section: Laguerre ˇ-Ensemble and The Exponential Toda Latticementioning

confidence: 99%

“…The measure with density exp(−β H AL ) and β > 0 is not bounded nor normalizable on the whole phase space. For this reason, we have defined the Gibbs ensemble as in (47). Furthermore, we observe that the measure (47) has a finite number of moments, which implies that the corresponding density of states of the Lax matrix (see (49) below), if it exists, would have a finite number of moments.…”

Section: Remarkmentioning

confidence: 99%

“…This is done through a one-to-one correspondence with the Laguerre β-ensemble at high temperature and with the antisymmetric Gaussian β-ensemble at high temperature, respectively. These are two known classes of random matrix ensembles, see [21,27,47] respectively. We consider other relevant cases of integrable systems, namely the focusing Ablowitz-Ladik lattice [1,2], the focusing Schur flow, and a class of integrable generalization of the Volterra lattice to short range interactions, called the Itoh-Narita-Bogoyavleskii (INB) additive and multiplicative lattices [17].…”

Section: Introductionmentioning

confidence: 99%

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Discrete Integrable Systems and Random Lax Matrices

et al. 2022

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We study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number N of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian $$\beta $$ β -ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz–Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states.

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Section: Parameter Limitmentioning

confidence: 98%

Section: Laguerre ˇ-Ensemble and The Exponential Toda Latticementioning

confidence: 99%

“…In order to study the density of states of the exponential Toda lattice, we need the following result proved in [47]. (21).…”

Section: Laguerre ˇ-Ensemble and The Exponential Toda Latticementioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discrete Integrable Systems and Random Lax Matrices

et al. 2022

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CLT for $$\beta $$-Ensembles at High Temperature and for Integrable Systems: A Transfer Operator Approach

Mazzuca,

Memin

2024

Ann. Henri Poincaré

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In this paper, we prove a polynomial central limit theorem for several integrable models and for the $$\beta $$ β -ensembles at high temperature with polynomial potential. Furthermore, we connect the mean values, the variances and the correlations of the moments of the Lax matrices of these integrable systems with the ones of the $$\beta $$ β -ensembles. Moreover, we show that the local functions’ space-correlations decay exponentially fast for the considered integrable systems. For these models, we also established a Berry–Esseen-type bound.

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Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular $$\beta $$-Ensemble and Double Confluent Heun Equation

Mazzuca

Грава

2023

Commun. Math. Phys.

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We consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, which is called defocusing Ablowitz–Ladik lattice. We consider periodic boundary conditions with period N and initial data sampled according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz–Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular $$\beta $$ β -ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz–Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz–Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation.

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On the mean density of states of some matrices related to the beta ensembles and an application to the Toda lattice

Cited by 9 publications

References 31 publications

Discrete Integrable Systems and Random Lax Matrices

Discrete Integrable Systems and Random Lax Matrices

CLT for $$\beta $$-Ensembles at High Temperature and for Integrable Systems: A Transfer Operator Approach

Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular $$\beta $$-Ensemble and Double Confluent Heun Equation

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