arXiv subjects

Guilherme L. F. Silva

Publications and source records attributed to Guilherme L. F. Silva.

11 recordsLinked to original sources

arXiv2026

The stochastic six vertex model and discrete orthogonal polynomial ensembles

Stochastic growth models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit remarkable fluctuation phenomena. While a variety of powerful methods have led to a detailed understanding of their typical fluctuations or large deviations, much less is known about behavior on intermediate, or moderate deviation, scales. Addressing this problem requires refined asymptotic control of the integrable structures underlying KPZ models. Motivated by this perspective, we study multiplicative statistics of discrete orthogonal polynomial ensembles (dOPEs) in different scaling regimes, with a particular focus on applications to tail probabilities of the height function in the stochastic six-vertex model. For a large class of dOPEs, we obtain robust singular asymptotic estimates for multiplicative statistics critically scaled near a saturated-to-band transition. These asymptotics exhibit universal crossover behavior, interpolating between Airy, Painlevé II, and Bessel-type regimes. Our proofs employ the Riemann-Hilbert Problem (RHP) approach to obtain asymptotics for the correlation kernel of a deformed version of the dOPE across the critical scaling windows. These asymptotics are then used on a double integral formula relating this kernel to partition function ratios, which may be of independent interest. At the technical level, the RHP analysis employs a novel parameter-dependent local parametrix, which requires a separate asymptotic analysis of its own. Using these results, together with a known identity relating a Laplace-type transform of the stochastic six-vertex model height function to a multiplicative statistic of the Meixner point process, we derive moderate deviation estimates for the height function in both the upper and lower tail regimes, with sharp exponents and constants.

math-ph↗

arXiv2026

Conditional thinning and multiplicative statistics of Laguerre-type orthogonal polynomial ensembles

We study the local statistics of orthogonal polynomial ensembles near a hard edge, subject to a multiplicative deformation of the measure. Probabilistically, this deformation corresponds to a position-dependent conditional thinning of the particles. We prove that, under critical hard edge scaling and for a large class of potentials and deformation symbols, the correlation kernel of the conditional ensemble converges to a universal limit, which we identify as the conditional thinned Bessel point process. We derive an explicit expression for this limiting kernel in terms of the solution to a nonlocal integrable system depending on a parameter. For a special choice of the parameter, this system was recently identified in the study of multiplicative statistics of the Bessel point process. Our results establish that this system governs the full correlation structure of the conditional Bessel point process, extending the classical connection between the standard Bessel kernel and the Painlevé V equation.

math-ph↗

arXiv2025

Deformations of biorthogonal ensembles and universality

We consider a large class of deformations of continuous and discrete biorthogonal ensembles and investigate their behavior in the limit of a large number of particles. We provide sufficient conditions to ensure that if a biorthogonal ensemble converges to a (universal) limiting process, then the deformed biorthogonal ensemble converges to a deformed version of the same limiting process. To construct the deformed version of the limiting process, we rely on a procedure of marking and conditioning. Our approach is based on an analysis of the probability generating functionals of the ensembles and is conceptually different from the traditional approach via correlation kernels. Thanks to this method, our sufficient conditions are rather mild and do not rely on much regularity of the original ensemble and of the deformation. As a consequence of our results, we obtain probabilistic interpretations of several Painlevé-type kernels that have been constructed in the literature, as deformations of classical sine and Airy point processes.

math.PR↗

arXiv2025

Deformations of OP ensembles in a bulk critical scaling

We study orthogonal polynomial ensembles whose weights are deformations of exponential weights, in the limit of a large number of particles. The deformation symbols we consider affect local fluctuations of the ensemble around a bulk point of the limiting spectrum. We identify the limiting kernel in terms of a solution to an integrable non-local differential equation. This novel kernel is the correlation kernel of a conditional thinned process starting from the Sine point process, and it is also related to a finite temperature deformation of the Sine kernel as recently studied by Claeys and Tarricone. We also unravel the effect of the deformation on the recurrence coefficients of the associated orthogonal polynomials, which display oscillatory behavior even in a one-cut regular situation for the limiting spectrum.

math-ph↗

arXiv2024

A Central Limit Theorem for intransitive dice

Intransitive dice $D^{(1)}, \ldots, D^{(\ell)}$ are dice such that $D^{(1)}$ has advantage when played against $D^{(2)}$, dice $D^{(2)}$ has advantage when played against $D^{(3)}$ and so on, up to $D^{(\ell)}$, which has advantage over $D^{(1)}$. In this twofold work, we first present (deterministic) results on the existence of general intransitive dice. Second and mainly, a central limit theorem for the vector of normalized victories of a die against the next one in the list when the faces of a die are i.i.d.\ random variables and all dice are independent, but different dice may have distinct distributions associated with them, as well as they may have distinct numbers of faces. Exploiting this central limit theorem, we derive two major consequences. First, we are able to obtain first order exponential asymptotics for the number of $\ell$-tuples of intransitive dice, when the number of faces of the dice grows. Second, we obtain a criterion to ensure that the asymptotic probability of observing intransitive dice is null, which applies to many cases, including all continuous distributions and many discrete ones.

math.PR↗

arXiv2022

Differential equations for the KPZ and periodic KPZ fixed points

The KPZ fixed point is a 2d random field, conjectured to be the universal limiting fluctuation field for the height function of models in the KPZ universality class. Similarly, the periodic KPZ fixed point is a conjectured universal field for spatially periodic models. For both fields, their multi-point distributions in the space-time domain have been computed recently. We show that for the case of the narrow-wedge initial condition, these multi-point distributions can be expressed in terms of so-called integrable operators. We then consider a class of operators that include the ones arising from the KPZ and the periodic KPZ fixed points, and find that they are related to various matrix integrable differential equations such as coupled matrix mKdV equations, coupled matrix NLS equations with complex time, and matrix KP-II equations. When applied to the KPZ fixed points, our results extend previously known differential equations for one-point distributions and equal-time, multi-position distributions to multi-time, multi-position setup.

math.PR↗

arXiv2022

Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix limit several associated quantities converge to limits which are universal in both the potential and the family of multiplicative statistics considered. In turn, such universal limits are described by the integro-differential Painlevé II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the KPZ equation at any finite time.

math-ph↗

arXiv2020

Spectral curves, variational problems and the hermitian matrix model with external source

We consider the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. All such models studied so far have a common feature: an associated cubic equation (spectral curve), one of whose solutions can be expressed in terms of the Cauchy transform of the limiting eigenvalue distribution. This is our starting point: we show that to any such spectral curve (not necessarily given by a random matrix ensemble) it corresponds a unique vector-valued measure with three components on the complex plane, characterized as a solution of a variational problem stated in terms of their logarithmic energy. We describe all possible geometries of the supports of these measures; in particular the third component, if non-trivial, lives on a contour on the plane. This general result is applied to the random matrix model with external source, under an additional assumption of uniform boundedness of the zeros of a sequence of average characteristic polynomials in the large matrix limit. It is shown that any limiting zero distribution for such a sequence can be written in terms of a solution of a spectral curve, and thus admits the variational description obtained in the first part of the paper. As a consequence of our analysis, we obtain that the density of this limiting measure can have only a handful of local behaviors: sine, Airy and their higher order type behavior, Pearcey or yet the fifth power of the cubic (but no higher order cubics can appear). We also compare our findings with the most general results available in the literature, showing that once an additional symmetry is imposed, our vector critical measure contains enough information to recover the solutions to the constrained equilibrium problem that was known to describe the limiting eigenvalue distribution in this symmetric situation.

math-ph↗

arXiv2020

Limiting one-point distribution of periodic TASEP

The relaxation time limit of the one-point distribution of the spatially periodic totally asymmetric simple exclusion process is expected to be the universal one point distribution for the models in the KPZ universality class in a periodic domain. Unlike the infinite line case, the limiting one point distribution depends non-trivially on the scaled time parameter. We study several properties of this distribution for the case of the periodic step and flat initial conditions. We show that the distribution changes from a Tracy-Widom distribution in the small time limit to the Gaussian distribution in the large time limit, and also obtain right tail estimate for all time. Furthermore, we establish a connection to integrable differential equations such as the KP equation, coupled systems of mKdV and nonlinear heat equations, and the KdV equation.

math.PR↗

arXiv2020

Supercritical Regime for the Kissing Polynomials

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function $e^{niλz}$ on $[-1,1]$, where $λ$ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when $λ$ is smaller than a certain critical value, $λ_c$. Our main goal is to compute their asymptotics when $λ>λ_c$. We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann-Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift-Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane. Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type. In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of $λ$ for which one cannot solve the model Riemann-Hilbert problem, and as such the corresponding polynomials fail to exist.

math.CA↗

arXiv2019

Large n limit for the product of two coupled random matrices

For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem. This vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure. We carry out the steepest descent analysis for a 4 $\times$ 4 matrix-valued Riemann-Hilbert problem, which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified Bessel functions. A careful study of the vector equilibrium problem, combined with this asymptotic analysis, ultimately leads to the aforementioned convergence result for the limiting mean distribution, an explicit form of the associated spectral curve, as well as local Sine, Meijer-G and Airy universality results for the squared singular values considered.

math-ph↗