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Abdelmalek Abdesselam

Publications and source records attributed to Abdelmalek Abdesselam.

At least 37 records · Page 2Linked to original sources
arXiv2014

Tridiagonalized GUE matrices are a matrix model for labeled mobiles

It is well-known that the number of planar maps with prescribed vertex degree distribution and suitable labeling can be represented as the leading coefficient of the $\frac{1}{N}$-expansion of a joint cumulant of traces of powers of an $N$-by-$N$ GUE matrix. Here we undertake the calculation of this leading coefficient in a different way. Firstly, we tridiagonalize the GUE matrix in the manner of Trotter and Dumitriu-Edelman and then alter it by conjugation to make the subdiagonal identically equal to $1$. Secondly, we apply the cluster expansion technique (specifically, the Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical mechanics. Thirdly, by sorting through the terms of the expansion thus generated we arrive at an alternate interpretation for the leading coefficient related to factorizations of the long cycle $(12\cdots n)\in S_n$. Finally, we reconcile the group-theoretical objects emerging from our calculation with the labeled mobiles of Bouttier-Di Francesco-Guitter.

math.PR↗
arXiv2013

QFT, RG, and all that, for mathematicians, in eleven pages

We present a quick introduction to quantum field theory and Wilson's theory of the renormalization group from the point of view of mathematical analysis. The presentation is geared primarily towards a probability theory, harmonic analysis and dynamical systems theory audience.

math.PR↗
arXiv2013

Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions

In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional p-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. G. Wilson. We also prove a mild form of universality for the model under consideration. Our main innovation is that our rigourous renormalization group formalism allows for space dependent couplings. We derive the relationship between mixed correlations and the dynamical systems features of our extended renormalization group transformation at a nontrivial fixed point. The key to our control of the composite field is a partial linearization theorem which is an infinite-dimensional version of the Koenigs Theorem in holomorphic dynamics. This is akin to a nonperturbative construction of a nonlinear scaling field in the sense of F. J. Wegner infinitesimally near the critical surface. Our presentation is essentially self-contained and geared towards a wider audience. While primarily concerning the areas of probability and mathematical physics we believe this article will be of interest to researchers in dynamical systems theory, harmonic analysis and number theory. It can also be profitably read by graduate students in theoretical physics with a craving for mathematical precision while struggling to learn the renormalization group.

math.PR↗
arXiv2012

Rigorous quantum field theory functional integrals over the p-adics: research announcement

In this short note we announce the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional p-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by Kenneth G. Wilson. We also prove a mild form of universality for the model under consideration.

math.PR↗
arXiv2012

Counting colored planar maps free-probabilistically

Our main result is an explicit operator-theoretic formula for the number of colored planar maps with a fixed set of stars each of which has a fixed set of half-edges with fixed coloration. The formula bounds the number of such colored planar maps well enough to prove convergence near the origin of generating functions arising naturally in the matrix model context. Such convergence is known but the proof of convergence proceeding by way of our main result is relatively simple. Besides Voiculescu's generalization of Wigner's semicircle law, our main technical tool is an integration identity representing the joint cumulant of several functions of a Gaussian random vector. The latter identity in the case of cumulants of order 2 reduces to one well-known as a means to prove the Poincare inequality. We derive the identity by combining the heat equation with the so-called BKAR formula from constructive quantum field theory and rigorous statistical mechanics.

math.PR↗
arXiv2012

On Hilbert covariants

Let F denote a binary form of order d over the complex numbers. If r is a divisor of d, then the Hilbert covariant H_{r,d}(F) vanishes exactly when F is the perfect power of an order r form. In geometric terms, the coefficients of H give defining equations for the image variety X of an embedding P^r->P^d. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on X. We prove that the ideal generated by the coefficients of H defines X as a scheme. Finally, we exhibit a generalisation of the Göttingen covariants to n-ary forms using the classical Clebsch transfer principle.

math.AG↗
arXiv2011

Analyticity of The Ground State Energy For Massless Nelson Models

We show that the ground state energy of the translationally invariant Nelson model, describing a particle coupled to a relativistic field of massless bosons, is an analytic function of the coupling constant and the total momentum. We derive an explicit expression for the ground state energy which is used to determine the effective mass.

math-ph↗
arXiv2010

The Ground State Energy of The Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated.

math-ph↗
arXiv2009

Renormalization group trajectories between two fixed points

We report on our recent rigorous construction of complete renormalization group trajectories between two fixed points for the three-dimensional phi-four model with modified propagator considered by Brydges, Mitter and Scoppola (BMS). These are discrete critical trajectories which connect the ultraviolet Gaussian fixed point to the nontrivial BMS infrared fixed point which is an analogue of the Wilson--Fisher fixed point. The renormalization group map is defined rigorously and nonperturbatively, without using the hierarchical approximation. The trajectories are constructed by a fixed point argument in a suitable Banach space of sequences, where one perturbs a nonlinear one-dimensional iteration.

math-ph↗
arXiv2009

On the volume conjecture for classical spin networks

We prove an upper bound for the evaluation of all classical SU(2) spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

math.GT↗
arXiv2009

Clustering Bounds on N-Point Correlations for Unbounded Spin Systems

We prove clustering estimates for the truncated correlations, i.e., cumulants of an unbounded spin system on the lattice. We provide a unified treatment, based on cluster expansion techniques, of four different regimes: large mass, small interaction between sites, large self-interaction, as well as the more delicate small self-interaction or `low temperature' regime. A clustering estimate in the latter regime is needed for the Bosonic case of the recent result obtained by Lukkarinen and Spohn on the rigorous control on kinetic scales of quantum fluids.

math-ph↗
arXiv2008

On the Linear Combinants of a Binary Pencil

Let A,B denote binary forms of order d, and let C_{2r-1} = (A,B)_{2r-1} be the sequence of their linear combinants for r between 1 and (d+1)/2. It is known that C_1 and C_3 together determine the pencil generated by A and B, and hence indirectly the higher C_{2r-1}. In this paper we exhibit explicit formulae for all r>2, which allow us to recover C_{2r-1} from the knowledge of C_1 and C_3. The calculations make use of the symbolic method of classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the second exterior power representation of S_d, for the group SL_2. We give an example for the group SL_3 to show that such a result may hold for other categories of representations.

math.AG↗
arXiv2008

The Higher Transvectants are Redundant

Let A, B denote generic binary forms, and let u_r = (A,B)_r denote their r-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the u_r. As a consequence, we show that each of the higher transvectants u_r, r>1, is redundant in the sense that it can be completely recovered from u_0 and u_1. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of SL_2-representations, and the notion of a 9-j symbol from the quantum theory of angular momentum. We give explicit computational examples for SL_3, g_2 and S_5 to show that this result has possible analogues for other categories of representations.

math.AG↗
arXiv2007

A Complete Renormalization Group Trajectory Between Two Fixed Points

We give a rigorous nonperturbative construction of a massless discrete trajectory for Wilson's exact renormalization group. The model is a three dimensional Euclidean field theory with a modified free propagator. The trajectory realizes the mean field to critical crossover from the ultraviolet Gaussian fixed point to an analog recently constructed by Brydges, Mitter and Scoppola of the Wilson-Fisher nontrivial fixed point.

math-ph↗
arXiv2006

On the Jacobian ideal of the binary discriminant

Let $Δ$ denote the discriminant of a generic binary $d$-ic. We show that for $d \ge 3$, the Jacobian ideal of $Δ$ is perfect of height 2. Moreover, we describe its SL_2-equivariant minimal resolution, and the associated invariant differential equations satisfied by $Δ$. A similar result is proved for the resultant of two forms of orders $d,e$, whenever $d \ge e-1$. We also explain the role of the Morley form in the determinantal formula for the resultant; this relies upon a calculation which is done in the appendix by A. Abdesselam.

math.AG↗
arXiv2005

A Computational Solution to a Question by Beauville on the Invariants of the Binary Quintic

We obtain an alternate proof of an injectivity result by Beauville for a map from the moduli space of quartic del Pezzo surfaces to the set of conjugacy classes of certain subgroups of the Cremona group. This amounts to showing that a projective configuration of five distinct unordered points on the line can be reconstructed from its five projective four-point subconfigurations. This is done by reduction to a question in the classical invariant theory of the binary quintic, which is solved by computer-assisted methods. More precisely, we show that six specific invariants of degree 24, the construction of which was explained to us by Beauville, generate all invariants the degrees of which are divisible by 48.

math.AC↗
arXiv2005

On the Wronskian combinants of binary forms

For generic binary forms $A_1,...,A_r$ of order $d$ we construct a class of combinants $C = \{\C_q: 0 \le q \le r, q \neq 1\}$, to be called the Wronskian combinants of the $A_i$. We show that the collection $C$ gives a projective imbedding of the Grassmannian $G(r,S_d)$, and as a corollary, any other combinant admits a formula as an iterated transvectant in the $C$. Our second main result characterizes those collections of binary forms which can arise as Wronskian combinants. These collections are the ones such that an associated algebraic differential equation has the maximal number of linearly independent polynomial solutions. Along the way we deduce some identities which connect Wronskians with transvectants.

math.AG↗