arXiv subjects

Guido Mazzuca

Publications and source records attributed to Guido Mazzuca.

14 recordsLinked to original sources

arXiv2026

Painlev\'e Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schr\"{o}dinger Equation with randomness

We establish universality for extremal solutions of the focusing nonlinear Schr\"{o}dinger equation. Extremal solutions are $N$-soliton solutions that achieve the theoretical maximal amplitude and diverge as $N \to \infty$. We consider extremal solutions with the discrete eigenvalues randomly drawn from sub-exponential distributions, and identify two distinct universality classes, determined by the macroscopic structure of the spectrum: the Painlev\'e--III rogue-wave solution, where the eigenvalues take the form $\lambda_j = v_j + i \mu_j$, and the Painlev\'e--V rogue wave solution, where $\lambda_j = -\zeta \, j + v_j + i \mu_j$, with $0 < \zeta < 1$. (In both cases, $\mu_{j}$ and $v_{j}$ are subexponential random variables.) Universality can then be summarized as follows: independently of the specific distribution of the eigenvalues, the rescaled solutions converge locally to a deterministic profile governed by the Painlev\'e-III equation in the first regime, and the Painlev\'e-V equation in the second. These results demonstrate that the formation of Painlev\'e-type rogue waves is a universal phenomenon robust to randomness.

math.AP↗

arXiv2026

$q$-deformation of the Marchenko-Pastur law

We study a $q$-deformed random unitary ensemble associated with the little-$q$ Laguerre weight, which provides a discrete analogue of the classical Laguerre unitary ensemble. In the double scaling regime $q=e^{-\lambda/N}$, where $N$ is the system size and $\lambda \ge 0$, we derive the limiting spectral distribution as $N\to \infty$, which yields a $q$-deformation of the Marchenko-Pastur law. The limiting density undergoes a phase transition at an explicitly determined critical value $\lambda_c$: for $\lambda<\lambda_c$, the support consists of a single band region, whereas for $\lambda>\lambda_c$ an additional saturated region emerges adjacent to the band region. Our derivation of the limiting distribution is based on three complementary approaches: the method of moments, the analysis of a constrained equilibrium problem, and the asymptotic zero distribution of orthogonal polynomials. As a consequence, we establish the convergence of the empirical measure as well as a large deviation principle. In addition, we derive closed-form expressions for the spectral moments using the combinatorial structure of orthogonal polynomials, and obtain large-$N$ expansions for these moments.

math.PR↗

arXiv2025

Soliton Synchronization with Randomness: Rogue Waves and Universality

We consider an $N$-soliton solution of the focusing nonlinear Schrödinger equations. We give conditions for the synchronous collision of these $N$ solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the $\operatorname{sinc}(x)$ function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. Namely the central collision peak exhibits universality: its spatial profile converges to the $\operatorname{sinc}(x)$ function, independently of the distribution. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-regime.

nlin.PS↗

arXiv2025

Discrete and Continuous Muttalib--Borodin Process: Large Deviations and Limit Shape Analysis

In this paper, we study the asymptotic behaviour of plane partitions distributed according to a $q^{\text{Volume}}$-weighted Muttalib--Borodin ensemble and its associated discrete point process. We establish a Large Deviation Principle for the process, explicitly characterizing the rate function. A defining feature of our model is the emergence of a strict upper bound on the macroscopic particle density, which translates the asymptotic analysis into a non-trivial constrained minimization problem. Through a rigorous Riemann--Hilbert analysis, we derive exact, closed-form formulas for the limit shape of the partitions across all parameter regimes. To the best of our knowledge, this represents the first time a constrained Riemann--Hilbert problem has been formulated and analytically solved for a bi-orthogonal ensemble. Our analysis allows to track the system through a macroscopic phase transition, computing the minimizer in both the subcritical and supercritical regimes. As a byproduct of our analysis, we obtain an explicit expression for the arctic curve that separates the ``frozen'' and ``liquid'' regions of the limit shape. Furthermore, we reveal that the equilibrium measure exhibits a continuously varying exponent at the hard edge departing from the universal fixed exponents typically observed in classical random matrix theory.

math.PR↗

arXiv2025

The formation of a soliton gas condensate for the focusing Nonlinear Schr\"odinger equation

In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schr\"{o}dinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which exhibit the formation (as $N \to \infty$) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov - Shabat operator for the NLS equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from $0$, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of $(x,t)$. We then consider more complex solutions with an extra soliton component, and provide rigorous justification of the predictions of the kinetic theory of solitons in this deterministic setting. This is to be distinguished from previous analyses of soliton gasses where the norming constants were tending to zero with $N$, and the asymptotic description only included elliptic waves in the long-time asymptotics.

math.AP↗

arXiv2024

Generalized Hydrodynamics for the Volterra lattice: Ballistic and nonballistic behavior of correlation functions

In recent years, a lot of effort has been put in describing the hydrodynamic behavior of integrable systems. In this paper, we describe such picture for the Volterra lattice. Specifically, we are able to explicitly compute the susceptibility matrix and the current-field correlation matrix in terms of the density of states of the Volterra lattice endowed with a Generalized Gibbs ensemble. Furthermore, we apply the theory of linear Generalized Hydrodynamics to describe the Euler scale behavior of the correlation functions. We anticipate that the solution to the Generalized Hydrodynamics equations develops shocks at $ξ_0=\frac{x}{t}$; so this linear approximation does not fully describe the behavior of correlation functions. Intrigued but this fact, we performed several numerical investigations which show that, exactly when the solution to the hydrodynamic equations develops shock, the correlation functions show an highly oscillatory behavior. In view of this empirical observation, we believe that at this point $ξ_0$ the diffusive contribution are not sub-leading corrections to the ballistic transport, but they are of the same order.

math-ph↗

arXiv2023

CLT for $β$ ensembles at high-temperature, and for integrable systems: a transfer operator approach

In this paper, we prove a polynomial Central Limit Theorem for several integrable models, and for the $β$-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 $β$-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.

math.PR↗

arXiv2023

Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $β$-ensemble and double confluent Heun equation

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 sample 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 $β$-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.

cond-mat.stat-mech↗

arXiv2023

Discrete integrable systems and random Lax matrices

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 $β$-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.

math-ph↗

arXiv2023

Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale

We report on molecular dynamics simulations of spacetime correlations of the Toda lattice in thermal equilibrium. The correlations of stretch, momentum, and energy are computed numerically over a wide range of pressure and temperature. Our numerical results are compared with the predictions from linearized generalized hydrodynamics on the Euler scale. The system size is N=3000,4000 and time t =600, at which ballistic scaling is well confirmed. With no adjustable parameters, the numerically obtained scaling functions agree with the theory within a precision of less than 3.5%.

cond-mat.stat-mech↗

arXiv2022

Large Deviations for Ablowitz-Ladik lattice, and the Schur flow

We consider the Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, and the Schur flow. We derive large deviations principles for the distribution of the empirical measures of the equilibrium measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limit in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble respectively.

math.PR↗

arXiv2021

The classical $β$-ensembles with $β$ proportional to $1/N$: from loop equations to Dyson's disordered chain

In the classical $β$-ensembles of random matrix theory, setting $β= 2 α/N$ and taking the $N \to \infty$ limit gives a statistical state depending on $α$. Using the loop equations for the classical $β$-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading $1/N$ correction to the density. From earlier literature the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying in particular the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterisation of the moments and covariances of monomial linear statistics is through recurrence relations. Also, we extend recent work which begins with the $β$-ensembles in the high temperature limit and constructs a family of tridiagonal matrices referred to as $α$-ensembles, obtaining a random anti-symmetric tridiagonal matrix with i.i.d.~gamma distributed random variables. From this we are able to supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain.

math.PR↗

arXiv2021

On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice

In this manuscript we study tridiagonal random matrix models related to the classical $β$-ensembles (Gaussian, Laguerre, Jacobi) in the high temperature regime, i.e. when the size $N$ of the matrix tends to infinity with the constraint that $βN=2α$ constant, $α> 0$. We call these ensembles the Gaussian, Laguerre and Jacobi $α$-ensembles and we prove the convergence of their empirical spectral distributions to their mean densities of states and we compute them explicitly. As an application we explicitly compute the mean density of states of the Lax matrix of the Toda lattice with periodic boundary conditions with respect to the Gibbs ensemble.

math.SP↗

arXiv2020

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate $t^{-\frac{1}{3}}$ for position and momentum correlations and as $t^{-\frac{2}{3}}$ for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate $t^{-\frac{1}{4}}$ for position and momentum correlators and with rate $t^{-\frac{1}{2}}$ for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.

math-ph↗