This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over 2-dimensional quantum gravity up to algebraic geometry with explicit computation of intersection numbers on the moduli space of complex curves.
The Kontsevich model, which has proved the Witten conjecture, is the simplest example of a matrix field theory. Generalisations of this model will be studied, where different potentials and the spectral dimension (defined by the asymptotics of the external matrix) are introduced. Because they are naturally embedded into a Riemann surface, the correlation functions are graded by the genus and the number of boundary components. The renormalisation procedure of quantum field theory leads to finite UV-limit.
We provide a method to determine closed Schwinger-Dyson equations with the usage of Ward-Takahashi identities in the continuum limit. The cubic (Kontsevich model) and the quartic (Grosse-Wulkenhaar model) potentials are studied separately. For the cubic potential, we show that the renormalisation procedure is compatible with topological recursion (TR). This means that the exact results computed by TR coincide perturbatively with the graph expansion renormalised by Zimmermann's forest formula. For the quartic model, the first correlation function (2-point function) is computed exactly. We give hints that the quartic model has structurally the same properties as the hermitian 2-matrix model with genus zero spectral curve.
We give a summary of the recent progress made by the authors and collaborators on the asymptotic analysis of the two matrix model with a quartic potential. The paper also contains a list of open problems.
We consider the problem of quantization of the bosonic membrane via the large $N$ limit of its matrix regularizations $H_N$ in Fock space. We prove that there exists a choice of the Fock space frequency such that $ H_N$ can be written as a sum of a non-interacting Hamiltonian $H_{0,N}$ and the original normal ordered quartic potential. Using this decomposition we obtain upper and lower bounds for the ground state energy in the planar limit, we study a perturbative expansion about the spectrum of $H_{0,N}$, and show that the spectral gap remains finite at $N=\infty$ at least up to the second or…
In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with a quartic monomial and a general even polynomial potential. We studied the correlation kernel for the eigenvalues of one of the matrices in asymptotic limit. We extended the results of Duits and Kuijlaars to the case when the limiting eigenvalue density for one of the matrices is supported on multiple intervals. The results are achieved by constructing the parametrix to a Riemann-Hilbert problem obtained by Duits and Kuijlaars with theta functions and then showing that this parametrix is well-defined by st…
The counting of alternating tangles in terms of their crossing number, number of external legs and connected components is presented here in a unified framework using quantum field-theoretic methods applied to a matrix model of colored links. The overcounting related to topological equivalence of diagrams is removed by means of a renormalization scheme of the matrix model; the corresponding ``renormalization equations'' are derived. Some particular cases are studied in detail and solved exactly.
We consider the two matrix model with an even quartic potential W(y)=y^4/4+alpha y^2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M_1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 x 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigen…
We use the Riemann-Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers $\mathscr{N}_j(g)$ of $4$-valent connected graphs with $j$ vertices on a compact Riemann surface of genus $g$. We explicitly evaluate these numbers for Riemann surfaces of genus $0,1,2,$ and $3$. Also, for a Riemann surface of an arbitrary genus $g$, we also calculate the leading term in the asymptotics of $\mathscr{N}_j(g)$ as the number of vertices tends to in…
Random tensor models are generalizations of random matrix models which admit $1/N$ expansions. In this article we show that the topological recursion, a modern approach to matrix models which solves the loop equations at all orders, is also satisfied in some tensor models. While it is obvious in some tensor models which are matrix models in disguise, it is far from clear that it can be applied to others. Here we focus on melonic interactions for which the models are best understood, and further restrict to the quartic case. Then Hubbard-Stratonovich transformation maps the tensor model to a mu…
Tensor models are measures for random tensors. They generalise matrix models and were developed to study random geometry in arbitrary dimension. Moreover, they are strongly connected to quantum gravity theories as additionally to the standard bare-bones models, they encompass the field theoretical approach to loop quantum gravity known as group field theory. In the present thesis, we focus on the restricted case of quartic tensor models, for which a far greater number of rigorous mathematical results have been proven. Quartic models can be re-written as multi-matrix models using the intermedia…
We derive bulk asymptotics of skew-orthogonal polynomials (sop) $π^{\bt}_{m}$, $β=1$, 4, defined w.r.t. the weight $\exp(-2NV(x))$, $V (x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ there exists an $ε> 0$, such that $ε\leq (m/N)\leq λ_{\rm cr}-ε$, where $λ_{\rm cr}$ is the critical value which separates sop with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between sop and orthogonal polynomials (op) and using asymptoti…
We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes with an exponent 1/2 in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a 4 x 4 Riemann-Hilbert problem related to the Hastings-McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appe…
We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}(Φ^3)$ is replaced by a quartic term $\mathrm{Tr}(Φ^4)$. Cumulants of the resulting measure are known to decompose into cycle types for which a recursive system of equations can be established. We develop a new, purely algebraic geometrical solution strategy for the two initial equations of the recursion, based on properties of Cauchy matrices. These structures led in subsequent work to the discovery that the quartic analogue of the Kontsevich model obeys blobbed topological recursion.
In this work several techniques to treat the partition function of the real scalar quartic quantum field theory on the Moyal plane is discussed. A factorisation approach requires the polytope volume for the diagonal subpolytope of symmetric stochastic matrices. This is determined asymptotically. The applicability of this method is tested for regime of weak coupling. Here the asymptotic character of the polytope volume alters the matrix structure of the original model leading to fictitious divergences.
We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and (q_{k,n})_k related to the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p_{n,n} as n tends to infinity. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t=0 reduces to the vector equilibrium problem tha…
After reviewing the Hermitian one matrix model, we will give a brief introduction to the Hermitian two matrix model and present a summary of some recent results on the asymptotic behavior of the two matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a $4\times 4$ Riemann-Hilbert problem related to Painlevé II equation. Also an open problem will be presented.
The analogue of Kontsevich's matrix Airy function, with the cubic potential $\operatorname{Tr}\big(Φ^3\big)$ replaced by a quartic term $\operatorname{Tr}\big(Φ^4\big)$ with the same covariance, provides a toy model for quantum field theory in which all correlation functions can be computed exactly and explicitly. In this paper we show that distinguished polynomials of correlation functions, themselves given by quickly growing series of Feynman ribbon graphs, sum up to much simpler and highly structured expressions. These expressions are deeply connected with meromorphic forms conjectured to o…
In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D-dimensional sphere, closely related to the "stacked" triangulations. For D<6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the D-dimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp cont…
This paper is devoted to the study of renormalization of the quartic melonic tensor model in dimension (=rank) five. We review the perturbative renormalization and the computation of the one loop beta function, confirming the asymptotic freedom of the model. We then define the Connes-Kreimer-like Hopf algebra describing the combinatorics of the renormalization of this model and we analyze in detail, at one- and two-loop levels, the Hochschild cohomology allowing to write the combinatorial Dyson-Schwinger equations. Feynman tensor graph Hopf subalgebras are also exhibited.
This paper is the first of a series aiming at proving rigorously the analyticity and the Borel summability of generic quartic bosonic and fermionic vector models (generalizing the O(N) vector model) in diverse dimensions. Both non-relativistic (Schrödinger) and relativistic (Klein-Gordon and Dirac) kinetic terms are considered. The 4-tensor defining the interactions is constant but otherwise arbitrary, up to the symmetries imposed by the statistics of the field. In this paper, we focus on models of low dimensions: bosons and fermions for d = 0, 1, and relativistic bosons for d = 2. Moreover, w…
We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the psi-function for the Hastings-McLeod solution to the Painlevé II equation. The proof is based on the Riemann-Hilbert approach.
