arXiv subjects

Masoud Khalkhali

Publications and source records attributed to Masoud Khalkhali.

At least 19 recordsLinked to original sources

arXiv2026

Asymmetric phase transitions in random noncommutative geometries

In this paper, we study the asymmetric phases of the quartic type (0, 1) and (1, 0) Dirac en- sembles via three approaches: the Riemann-Hilbert approach, bootstrapping with positivity, and Hamiltonian Monte Carlo (HMC) simulations. The focus of this work is on asymmetric solutions to the Schwinger-Dyson and saddle point equations of these models, whose solution spaces prove deeply intricate. Via the Riemann-Hilbert approach, we are able to give explicit formulae for the eigenvalue distributions and free energy of various solutions. Using Hamiltonian Monte Carlo simulations, we are able to reconstruct the phase structure. Lastly, using bootstrapping with positivity, we are able to reconstruct the eigenvalue distribution of these models from their bootstrapped moments. All three methods show excellent agreement for a large matrix size.

math-ph↗

arXiv2026

Mohsen Hachtroudi and the Geometry of Differential Equations

Mohsen Hachtroudi (1908--1976) was one of the founders of modern mathematics in Iran and the first Iranian mathematician to gain international recognition for original research in modern times. A student of Élie Cartan, he introduced the canonical projective connection now known as the \emph{Hachtroudi connection}, whose significance has become increasingly apparent through later developments in Cartan geometry, CR geometry, and parabolic geometry. This article combines a historical and mathematical study of\\ Hachtroudi's life and work. It examines his education in Iran and France, his role in establishing modern mathematical research and higher education in Iran, and his influence as a teacher, essayist, and public intellectual. It also presents a modern account of his mathematical contributions and their continuing impact on the geometry of differential equations and the theory of geometric structures.

math.HO↗

arXiv2026

The Schwinger-Dyson equations for random fuzzy geometries coupled to matter

In this work we study the Schwinger-Dyson equations and saddle point equations of matrix integrals that come from type $(0,1)$ random fuzzy geometries coupled to fermions or bosons. Such random fuzzy geometries are bi-tracial Hermitian matrix ensembles with a determinant contribution in the integrand. We derive the Schwinger-Dyson equations using complex analytic techniques from the saddle point equation. For arbitrary potentials with either bosonic or fermionic contributions, their Schwinger-Dyson equations can be solved iteratively. For both the Gaussian models with either one boson or fermion we rigorously derive the formula for the free energy and first moment in terms of elliptic integrals. In the bosonic case this solution is closely related to the Hoppe model and the three-colour model.

math-ph↗

arXiv2025

Bootstrapping Noncommutative Geometry with Dirac Ensembles

This paper surveys a bootstrap framework for random Dirac operators arising from finite spectral triples in noncommutative geometry. Motivated by a toy model for quantum gravity to replace integration over metrics by integration over Dirac operators, we give an overview of multitrace and multimatrix random matrix models built from spectral triples and analyze them in the large $N$ limit using positivity constraints on Hankel moment matrices. In this setting, the bootstrap philosophy, originating in the S-matrix program and revived in modern conformal bootstrap theory, reappears as a rigorous analytic tool for extracting spectral data from consistency alone, without solving the model explicitly. We explain how Schwinger-Dyson equations, factorization at large $N$, and the noncommutative moment problem lead to finite-dimensional semidefinite programs whose feasible regions encode the allowed pairs of coupling constants and moments. Connections with spectral geometry, in particular the study of Laplace eigenvalues, are also discussed, illustrating how bootstrapping provides a unified mechanism for deriving bounds in both commutative and noncommutative settings.

math-ph↗

arXiv2025

Noncommutative geometry on the Berkovich projective line

We construct several $C^*$-algebras and spectral triples associated to the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}({\mathbb{C}_p})$. In the commutative setting, we construct a spectral triple as a direct limit over finite $\mathbb{R}$-trees. More general $C^*$-algebras generated by partial isometries are also presented. We use their representations to associate a Perron-Frobenius operator and a family of projection valued measures. Finally, we show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of the crossed product algebra with a Schottky subgroup of $\mathrm{PGL}_2(\mathbb{C}_p)$.

math.FA↗

arXiv2025

Bootstrapping the critical behavior of multi-matrix models

Given a matrix model, by combining the Schwinger-Dyson equations with positivity constraints on its solutions, in the large $N$ limit one is able to obtain explicit and numerical bounds on its moments. This technique is known as bootstrapping with positivity. In this paper we use this technique to estimate the critical points and exponents of several multi-matrix models. As a proof of concept, we first show it can be used to find the well-studied quartic single matrix model's critical phenomena. We then apply the method to several similar ``unsolved" 2-matrix models with various quartic interactions. We conjecture and present strong evidence for the string susceptibility exponent for some of these models to be $γ= 1/2$, which heuristically indicates that the continuum limit will likely be the Continuum Random Tree. For the other 2-matrix models, we find estimates of new string susceptibility exponents that may indicate a new continuum limit. We then study an unsolved 3-matrix model that generalizes the 3-colour model with cubic interactions. Additionally, for all of these models, we are able to derive explicitly the first several terms of the free energy in the large $N$ limit as a power series expansion in the coupling constants at zero by exploiting the structure of the Schwinger-Dyson equations.

math-ph↗

arXiv2024

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

We compute the first twenty moments of three convergent quartic bi-tracial 2-matrix ensembles in the large $N$ limit. These ensembles are toy models for Euclidean quantum gravity originally proposed by John Barrett and collaborators. A perturbative solution is found for the first twenty moments using the Schwinger-Dyson equations and properties of certain bi-colored unstable maps associated to the model. We then apply a result of Guionnet et al. to show that the perturbative and convergent solution coincide for a small neighbourhood of the coupling constants. For each model we compute an explicit expression for the free energy, critical points, and critical exponents in the large $N$ limit. In particular, the string susceptibility is found to be $γ=1/2$, hinting that the associated universality class of the model is the continuous random tree.

math-ph↗

arXiv2024

Random Finite Noncommutative Geometries and Topological Recursion

In this paper we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples ${(\mathcal{A}, \mathcal{H}, D , γ, J) \,}$, called random matrix geometries of type ${(1,0) \,}$, with a fixed fermion space ${(\mathcal{A}, \mathcal{H}, γ, J) \,}$, and a distribution of the form ${e^{- \mathcal{S} (D)} {\mathop{}\!\mathrm{d}} D}$ over the moduli space of Dirac operators. The action functional ${\mathcal{S} (D)}$ is considered to be a sum of terms of the form ${\prod_{i=1}^s \mathrm{Tr} \left( {D^{n_i}} \right)}$ for arbitrary ${s \geqslant 1 \,}$. The Schwinger-Dyson equations satisfied by the connected correlators ${W_n}$ of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients ${W_{g,n}}$ of the large $N$ expansion of ${W_n}$'s enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve ${\left( {Σ, ω_{0,1} , ω_{0,2}} \right)}$ of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential ${ω_{0,2}}$ in terms of the formal parameters of the model.

math-ph↗

arXiv2024

Large N limit of fuzzy geometries coupled to fermions

In this paper we present an analysis of the large N limit of a family of quartic Dirac ensembles based on (0, 1) fuzzy geometries that are coupled to fermions. These Dirac ensembles are examples of single-matrix, multi-trace matrix ensembles. Additionally, they serve as examples of integer-valued $β$-ensembles. Convergence of the spectral density in the large N limit for a large class of such matrix ensembles is proven, improving on existing results. The main results of this paper are the addition of the fermionic contribution in the matrix ensemble and the investigation of spectral estimators for finite dimensional spectral triples

math-ph↗

arXiv2023

Double scaling limits of Dirac ensembles and Liouville quantum gravity

In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory.

math-ph↗

arXiv2023

A Selberg Trace Formula for $\text{GL}_{3}(\mathbb{F}_p)\backslash \text{GL}_{3}(\mathbb{F}_q)/K$

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group $\text{GL}_{3}(\mathbb{F}_q).$ By considering a cubic extension of the finite field $\mathbb{F}_q$, we define an analog of the upper half space and an action of $\text{GL}_{3}(\mathbb{F}_q)$ on it. To compute the orbital sums we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula we decompose the induced representation $ρ= \text{Ind}_Γ^{G} 1$ for $G= \text{GL}_{3}(\mathbb{F}_q) $ and $ Γ= \text{GL}_{3}(\mathbb{F}_p).$

math.NT↗

arXiv2022

From Noncommutative Geometry to Random Matrix Theory

We review recent progress in the analytic study of random matrix models suggested by noncommutative geometry. One considers fuzzy spectral triples where the space of possible Dirac operators is assigned a probability distribution. These ensembles of Dirac operators are constructed as toy models of Euclidean quantum gravity on finite noncommutative spaces and display many interesting properties. The ensembles exhibit spectral phase transitions, and near these phase transitions they show manifold-like behavior. In certain cases one can recover Liouville quantum gravity in the double scaling limit. We highlight examples where bootstrap techniques, Coulomb gas methods, and Topological Recursion are applicable.

hep-th↗

arXiv2022

Hopf Cyclic Cohomology and Beyond

This paper is an introduction to Hopf cyclic cohomology with an emphasis on its most recent developments. We cover three major areas: the original definition of Hopf cyclic cohomology by Connes and Moscovici as an outgrowth of their study of transverse index theory on foliated manifolds, the introduction of Hopf cyclic cohomology with coefficients by Hajac-Khalkhali-Rangipour-Sommerhauser, and finally the latest episode on unifying the coefficients as well as extending the notion to more general settings beyond Hopf algebras. In particular, the last section discusses the relative Hopf cyclic theory that arises in the braided monoidal category settings.

math.KT↗

arXiv2022

Spectral Statistics of Dirac Ensembles

In this paper we find spectral properties in the large $N$ limit of Dirac operators that come from random finite noncommutative geometries. In particular for a Gaussian potential the limiting eigenvalue spectrum is shown to be universal regardless of the geometry and is given by the convolution of the semicircle law with itself. For simple non-Gaussian models this convolution property is also evident. In order to prove these results we show that a wide class of multi-trace multimatrix models have a genus expansion.

hep-th↗

arXiv2022

Bootstrapping Dirac Ensembles

We apply the bootstrap technique to find the moments of certain multi-trace and multi-matrix random matrix models suggested by noncommutative geometry. Using bootstrapping we are able to find the relationships between the coupling constant of these models and their second moments. Using the Schwinger-Dyson equations, all other moments can be expressed in terms of the coupling constant and the second moment. Explicit relations for higher mixed moments are obtained.

hep-th↗

arXiv2020

Phase Transition in Random Noncommutative Geometries

We present an analytic proof of the existence of phase transition in the large $N$ limit of certain random noncommutaitve geometries. These geometries can be expressed as ensembles of Dirac operators. When they reduce to single matrix ensembles, one can apply the Coulomb gas method to find the empirical spectral distribution. We elaborate on the nature of the large $N$ spectral distribution of the Dirac operator itself. Furthermore, we show that these models exhibit both a single and double cut region for certain values of the order parameter and find the exact value where the transition occurs.

math-ph↗

arXiv2020

Second Quantization and the Spectral Action

We consider both the bosonic and fermionic second quantization of spectral triples in the presence of a chemical potential. We show that the von Neumann entropy and the average energy of the Gibbs state defined by the bosonic and fermionic grand partition function can be expressed as spectral actions. It turns out that all spectral action coefficients can be given in terms of the modified Bessel functions. In the fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $μ$ approaches $0,$ can be expressed in terms of the Riemann zeta function. This recovers a result of Chamseddine-Connes-van Suijlekom.

math-ph↗

arXiv2020

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.

math.QA↗