arXiv subjects

G. Vernizzi

Publications and source records attributed to G. Vernizzi.

16 recordsLinked to original sources

arXiv2019

Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane

We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials $C_n^{(1+α)}(z)$ for $α>-1$ containing the Legendre polynomials $P_n(z)$, and the subset $P_n^{(α+\frac12,\pm\frac12)}(z)$ of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of first up to fourth kind. The limit $α\to\infty$ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.

math-ph↗

arXiv2005

The Topology of Pseudoknotted Homopolymers

We consider the folding of a self-avoiding homopolymer on a lattice, with saturating hydrogen bond interactions. Our goal is to numerically evaluate the statistical distribution of the topological genus of pseudoknotted configurations. The genus has been recently proposed for classifying pseudoknots (and their topological complexity) in the context of RNA folding. We compare our results on the distribution of the genus of pseudoknots, with the theoretical predictions of an existing combinatorial model for an infinitely flexible and stretchable homopolymer. We thus obtain that steric and geometric constraints considerably limit the topological complexity of pseudoknotted configurations, as it occurs for instance in real RNA molecules. We also analyze the scaling properties at large homopolymer length, and the genus distributions above and below the critical temperature between the swollen phase and the compact-globule phase, both in two and three dimensions.

q-bio.BM↗

arXiv2004

Enumeration of RNA structures by Matrix Models

We enumerate the number of RNA contact structures according to their genus, i.e. the topological character of their pseudoknots. By using a recently proposed matrix model formulation for the RNA folding problem, we obtain exact results for the simple case of an RNA molecule with an infinitely flexible backbone, in which any arbitrary pair of bases is allowed. We analyze the distribution of the genus of pseudoknots as a function of the total number of nucleotides along the phosphate-sugar backbone.

q-bio.BM↗

arXiv2004

Prediction of RNA pseudoknots by Monte Carlo simulations

In this paper we consider the problem of RNA folding with pseudoknots. We use a graphical representation in which the secondary structures are described by planar diagrams. Pseudoknots are identified as non-planar diagrams. We analyze the non-planar topologies of RNA structures and propose a classification of RNA pseudoknots according to the minimal genus of the surface on which the RNA structure can be embedded. This classification provides a simple and natural way to tackle the problem of RNA folding prediction in presence of pseudoknots. Based on that approach, we describe a Monte Carlo algorithm for the prediction of pseudoknots in an RNA molecule.

q-bio.BM↗

arXiv2004

On matrix model partition functions for QCD with chemical potential

Partition functions of two different matrix models for QCD with chemical potential are computed for an arbitrary number of quark and complex conjugate anti-quark flavors. In the large-N limit of weak nonhermiticity complete agreement is found between the two models. This supports the universality of such fermionic partition functions, that is of products of characteristic polynomials in the complex plane. In the strong nonhermiticity limit agreement is found for an equal number of quark and conjugate flavours. For a general flavor content the equality of partition functions holds only for small chemical potential. The chiral phase transition is analyzed for an arbitrary number of quarks, where the free energy presents a discontinuity of first order at a critical chemical potential. In the case of nondegenerate flavors there is first order phase transition for each separate mass scale.

hep-th↗

arXiv2003

3D Lorentzian Quantum Gravity from the asymmetric ABAB matrix model

The asymmetric ABAB-matrix model describes the transfer matrix of three-dimensional Lorentzian quantum gravity. We study perturbatively the scaling of the ABAB-matrix model in the neighbourhood of its symmetric solution and deduce the associated renormalization of three-dimensional Lorentzian quantum gravity.

hep-th↗

arXiv2002

Characteristic Polynomials of Complex Random Matrix Models

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.

hep-th↗

arXiv2002

Rotational Symmetry Breaking in Multi-Matrix Models

We consider a class of multi-matrix models with an action which is O(D) invariant, where D is the number of NxN Hermitian matrices X_μ, μ=1,...,D. The action is a function of all the elementary symmetric functions of the matrix $T_{μν}=Tr(X_μX_ν)/N$. We address the issue whether the O(D) symmetry is spontaneously broken when the size N of the matrices goes to infinity. The phase diagram in the space of the parameters of the model reveals the existence of a critical boundary where the O(D) symmetry is maximally broken.

hep-th↗

arXiv2002

New Critical Matrix Models and Generalized Universality

We study a class of one-matrix models with an action containing nonpolynomial terms. By tuning the coupling constants in the action to criticality we obtain that the eigenvalue density vanishes as an arbitrary real power at the origin, thus defining a new class of multicritical matrix models. The corresponding microscopic scaling law is given and possible applications to the chiral phase transition in QCD are discussed. For generic coupling constants off-criticality we prove that all microscopic correlation functions at the origin of the spectrum remain in the known Bessel universality class. An arbitrary number of Dirac mass terms can be included and the corresponding massive universality is maintained as well. We also investigate the critical behavior at the edge of the spectrum: there, in contrast to the behavior at the origin, we find the same critical exponents as derived from matrix models with a polynomial action.

hep-th↗

arXiv2001

Yang-Mills Integrals

Two results are presented for reduced Yang-Mills integrals with different symmetry groups and dimensions: the first is a compact integral representation in terms of the relevant variables of the integral, the second is a method to analytically evaluate the integrals in cases of low order. This is exhibited by evaluating a Yang-Mills integral over real symmetric matrices of order 3.

hep-th↗

arXiv2001

Lorentzian 3d Gravity with Wormholes via Matrix Models

We uncover a surprising correspondence between a non-perturbative formulation of three-dimensional Lorentzian quantum gravity and a hermitian two-matrix model with ABAB-interaction. The gravitational transfer matrix can be expressed as the logarithm of a two-matrix integral, and we deduce from the known structure of the latter that the model has two phases. In the phase of weak gravity, well-defined two-dimensional universes propagate in proper time, whereas in the strong-coupling phase the spatial hypersurfaces disintegrate into many components connected by wormholes.

hep-th↗

arXiv2001

On the Microscopic Spectra of the Massive Dirac Operator for Chiral Orthogonal and Chiral Symplectic Ensembles

Using Random Matrix Theory we set out to compute the microscopic correlators of the Euclidean Dirac operator in four dimensions. In particular we consider: the chiral Orthogonal Ensemble (chOE), corresponding to a Yang-Mills theory with two colors and fermions in the fundamental representation, and the chiral Symplectic Ensemble (chSE), corresponding to any number of colors and fermions in the adjoint representation. In both cases we deal with an arbitrary number of massive fermions. We use a recent method proposed by H. Widom for deriving closed formulas for the scalar kernels from which all spectral correlation functions of the chGOE and chGSE can be determined. Moreover, we obtain complete analytic expressions of such correlators in the double microscopic limit, extending previously known results of four-dimensional QCD at beta=1 and beta=4 to the general case with N_f flavors, with arbitrary quark masses and arbitrary topological charge.

hep-th↗

arXiv2000

Macroscopic and Microscopic (Non-)Universality of Compact Support Random Matrix Theory

A random matrix model with a sigma-model like constraint, the restricted trace ensemble (RTE), is solved in the large-n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though non-universal expression for G(z,w), which extends recursively to all higher k-point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large-n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials $V(M)=M^{2p}$, we provide a relation valid for finite-n between the k-point correlation function of the RTE and the unconstrained model. In the microscopic large-n limit they coincide which proves the microscopic universality of RTEs.

hep-th↗

arXiv2000

Spontaneous Breakdown of Lorentz Invariance in IIB Matrix Model

We study the IIB matrix model, which is conjectured to be a nonperturbative definition of superstring theory, by introducing an integer deformation parameter `nu' which couples to the imaginary part of the effective action induced by fermions. The deformed IIB matrix model continues to be well-defined for arbitrary `nu', and it preserves gauge invariance, Lorentz invariance, and the cluster property. We study the model at `nu' = infinity using a saddle-point analysis, and show that ten-dimensional Lorentz invariance is spontaneously broken at least down to an eight-dimensional one. We argue that it is likely that the remaining eight-dimensional Lorentz invariance is further broken, which can be checked by integrating over the saddle-point configurations using standard Monte Carlo simulation.

hep-th↗

arXiv1999

Non-universality of compact support probability distributions in random matrix theory

The two-point resolvent is calculated in the large-n limit for the generalized fixed and bounded trace ensembles. It is shown to disagree with the one of the canonical Gaussian ensemble by a non-universal part which is given explicitly for all monomial potentials $V(M)=M^{2p}$. Moreover, we prove that for the generalized fixed and bounded trace ensemble all k-point resolvents agree in the large-n limit, despite their non-universality.

cond-mat↗

arXiv1998

Compact support probability distributions in random matrix theory

We consider a generalization of the fixed and bounded trace ensembles introduced by Bronk and Rosenzweig up to an arbitrary polynomial potential. In the large-N limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the "canonical" ensemble with measure exp[-$n$Tr V(M)]. The mapping of the corresponding phase boundaries is illuminated in an explicit example. In the case of a Gaussian potential we are able to derive exact expressions for the one- and two-point correlator for finite $n$, having finite support.

cond-mat↗