Sanjaye Ramgoolam
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Sanjaye Ramgoolam has a total of 57 co-authors. Here are the top co-authors by number of shared publications:
- Robert de Mello Koch
Huzhou University
23 shared publications - Adrian Padellaro
Faculty of Physics, Bielefeld University, PO-Box 100131, D-33501 Bielefeld, Germany
10 shared publications - Amihay Hanany
Imperial College London
9 shared publications - Jurgis Pasukonis
Department of Physics, Queen Mary, University of London, London, U.K.
9 shared publications - George Barnes
Centre for Theoretical Physics, School of Physical and Chemical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK
8 shared publications - Joseph Ben Geloun
Laboratoire d’Informatique de Paris Nord UMR CNRS 7030 Université Paris 13 1 , 99, Avenue J.-B. Clement, 93430 Villetaneuse, France
8 shared publications
Latest Publications
- Open Access
- 17 August 2026
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2026 (8) , 131
Critical dimensions and small cycle dominance from all-orders asymptotics of d-matrix theory
A bstract: Supersymmetric sectors of $$ \mathcal{N} $$ = 4 super-Yang-Mills theory motivate the study of the partition function for the counting of gauge-invariant functions of d = 2, 3 matrices transforming under the adjoint action of U ( N ). The partition function $$ {\mathcal{Z}}_d(x) $$ in the large N limit has a known Hagedorn phase transition at x = d −1 which provides a simple model for the phase structure of the thermal partition function of SYM. We study the all-orders asymptotic expansion of $$ {\mathcal{Z}}_d(x) $$ based on a geometric picture of concentric circles of poles in the complex plane accumulating in a natural boundary at | x | = 1. We find that the order by order structure has a precise combinatorial interpretation organized in terms of increasing cycle size of permutations arising in the enumeration of the invariants. We refer to this organization as small-cycle dominance, and find that it extends to refined versions of the partition functions depending on several complex variables. An analysis of the coefficients in the asymptotic expansion of $$ {\mathcal{Z}}_d(x) $$ using the modular property of the Dedekind eta function reveals that the asymptotic expansion is actually convergent for d ≥ d crit = 13. A fermionic version of $$ {\mathcal{Z}}_d(x) $$ has an analogous critical dimension of d crit = 7. This distinction indicates that the partition functions of the matrix models can be completely reconstructed from their high-energy (UV) limit for d ≥ d crit whereas additional input is required to reconstruct the exact coefficients of the low-energy (IR) expansion for 2 ≤ d ≤ d crit − 1.- 17 August 2026
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2026 (8) , 131
Open AccessCritical dimensions and small cycle dominance from all-orders asymptotics of d-matrix theory
A bstract: Supersymmetric sectors of $$ \mathcal{N} $$ = 4 super-Yang-Mills theory motivate the study of the partition function for the counting of gauge-invariant functions of d = 2, 3 matrices transforming under the adjoint action of U ( N ). The partition function $$ {\mathcal{Z}}_d(x) $$ in the large N limit has a known Hagedorn phase transition at x = d −1 which provides a simple model for the phase structure of the thermal partition function of SYM. We study the all-orders asymptotic expansion of $$ {\mathcal{Z}}_d(x) $$ based on a geometric picture of concentric circles of poles in the complex plane accumulating in a natural boundary at | x | = 1. We find that the order by order structure has a precise combinatorial interpretation organized in terms of increasing cycle size of permutations arising in the enumeration of the invariants. We refer to this organization as small-cycle dominance, and find that it extends to refined versions of the partition functions depending on several complex variables. An analysis of the coefficients in the asymptotic expansion of $$ {\mathcal{Z}}_d(x) $$ using the modular property of the Dedekind eta function reveals that the asymptotic expansion is actually convergent for d ≥ d crit = 13. A fermionic version of $$ {\mathcal{Z}}_d(x) $$ has an analogous critical dimension of d crit = 7. This distinction indicates that the partition functions of the matrix models can be completely reconstructed from their high-energy (UV) limit for d ≥ d crit whereas additional input is required to reconstruct the exact coefficients of the low-energy (IR) expansion for 2 ≤ d ≤ d crit − 1. - Open Access
- 1 June 2026
- Published by IOP Publishing in Machine Learning: Science and Technology
- Vol. 7 (3) , 035038
Approximate Gaussianity beyond initialisation in neural networks
Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.
- 1 June 2026
- Published by IOP Publishing in Machine Learning: Science and Technology
- Vol. 7 (3) , 035038
Open AccessApproximate Gaussianity beyond initialisation in neural networks
Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.
- Open Access
Finite-dimensional algebras, gauge-string duality and thermodynamics
Gauge-invariant polynomial functions of matrix and tensor variables capture combinatorial structures of gauge-string duality, which can be usefully organised using finite-dimensional associative algebras. I review recent work on eigenvalue systems using these algebras as state spaces, which provide efficient computational algorithms for the construction of orthogonal bases in the multi-matrix case. Algebraic counting formulae in matrix and tensor systems with $U(N)$ as well as $S_N$ symmetry have led to gauged quantum mechanical models which display a negative branch of specific heat capacity in the micro-canonical ensemble followed by positive specific heat capacity at larger energies measured by a polynomial degree parameter $n$. The negative branch is associated with near-exponential or factorial growth of degeneracies for $ n \gg 1$ in a region of large $N$ stability, while the positive branch occurs when the finite $N$ reduction of degrees of freedom takes over as $n$ becomes sufficiently large compared to $N$.Open AccessFinite-dimensional algebras, gauge-string duality and thermodynamics
Gauge-invariant polynomial functions of matrix and tensor variables capture combinatorial structures of gauge-string duality, which can be usefully organised using finite-dimensional associative algebras. I review recent work on eigenvalue systems using these algebras as state spaces, which provide efficient computational algorithms for the construction of orthogonal bases in the multi-matrix case. Algebraic counting formulae in matrix and tensor systems with $U(N)$ as well as $S_N$ symmetry have led to gauged quantum mechanical models which display a negative branch of specific heat capacity in the micro-canonical ensemble followed by positive specific heat capacity at larger energies measured by a polynomial degree parameter $n$. The negative branch is associated with near-exponential or factorial growth of degeneracies for $ n \gg 1$ in a region of large $N$ stability, while the positive branch occurs when the finite $N$ reduction of degrees of freedom takes over as $n$ becomes sufficiently large compared to $N$. - 15 December 2025
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 58 (50) , 505401
Row and column detection complexities of character tables
Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define a notion of generator complexity for commutative Frobenius algebras with combinatorial bases. In the context of finite groups, this gives rise to row and column versions of generator complexity for character tables. We investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.
- 15 December 2025
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 58 (50) , 505401
Row and column detection complexities of character tables
Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define a notion of generator complexity for commutative Frobenius algebras with combinatorial bases. In the context of finite groups, this gives rise to row and column versions of generator complexity for character tables. We investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.
- Open Access
Approximate Gaussianity Beyond Initialisation in Neural Networks
Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.
Open AccessApproximate Gaussianity Beyond Initialisation in Neural Networks
Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.
- Open Access
Row and column detection complexities of character tables
Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define row and column versions of detection complexities for character tables, and investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.
Open AccessRow and column detection complexities of character tables
Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define row and column versions of detection complexities for character tables, and investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.
- Open Access
- 18 February 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (2) , 1-73
Eigenvalue systems for integer orthogonal bases of multi-matrix invariants at finite N
Multi-matrix invariants, and in particular the scalar multi-trace operators of $$ \mathcal{N} $$ = 4 SYM with U(N) gauge symmetry, can be described using permutation centraliser algebras (PCA), which are generalisations of the symmetric group algebras and independent of N. Free-field two-point functions define an N-dependent inner product on the PCA, and bases of operators have been constructed which are orthogonal at finite N. Two such bases are well-known, the restricted Schur and covariant bases, and both definitions involve representation-theoretic quantities such as Young diagram labels, multiplicity labels, branching and Clebsch-Gordan coefficients for symmetric groups. The explicit computation of these coefficients grows rapidly in complexity as the operator length increases. We develop a new method for explicitly constructing all the operators with specified Young diagram labels, based on an N-independent integer eigensystem formulated in the PCA. The eigensystem construction naturally leads to orthogonal basis elements which are integer linear combinations of the multi-trace operators, and the N-dependence of their norms are simple known dimension factors. We provide examples and give computer codes in SageMath which efficiently implement the construction for operators of classical dimension up to 14. While the restricted Schur basis relies on the Artin-Wedderburn decomposition of symmetric group algebras, the covariant basis relies on a variant which we refer to as the Kronecker decomposition. Analogous decompositions exist for any finite group algebra and the eigenvalue construction of integer orthogonal bases extends to the group algebra of any finite group with rational characters.- 18 February 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (2) , 1-73
Open AccessEigenvalue systems for integer orthogonal bases of multi-matrix invariants at finite N
Multi-matrix invariants, and in particular the scalar multi-trace operators of $$ \mathcal{N} $$ = 4 SYM with U(N) gauge symmetry, can be described using permutation centraliser algebras (PCA), which are generalisations of the symmetric group algebras and independent of N. Free-field two-point functions define an N-dependent inner product on the PCA, and bases of operators have been constructed which are orthogonal at finite N. Two such bases are well-known, the restricted Schur and covariant bases, and both definitions involve representation-theoretic quantities such as Young diagram labels, multiplicity labels, branching and Clebsch-Gordan coefficients for symmetric groups. The explicit computation of these coefficients grows rapidly in complexity as the operator length increases. We develop a new method for explicitly constructing all the operators with specified Young diagram labels, based on an N-independent integer eigensystem formulated in the PCA. The eigensystem construction naturally leads to orthogonal basis elements which are integer linear combinations of the multi-trace operators, and the N-dependence of their norms are simple known dimension factors. We provide examples and give computer codes in SageMath which efficiently implement the construction for operators of classical dimension up to 14. While the restricted Schur basis relies on the Artin-Wedderburn decomposition of symmetric group algebras, the covariant basis relies on a variant which we refer to as the Kronecker decomposition. Analogous decompositions exist for any finite group algebra and the eigenvalue construction of integer orthogonal bases extends to the group algebra of any finite group with rational characters. - Open Access
- 20 December 2024
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2024 (12) , 1-67
Permutation invariant matrix quantum thermodynamics and negative specific heat capacities in large N systems
A bstract: We study the thermodynamic properties of the simplest gauged permutation invariant matrix quantum mechanical system of oscillators, for general matrix size N . In the canonical ensemble, the model has a transition at a temperature T given by $$ x={e}^{-1/T}\sim {x}_c={e}^{-1/{T}_c}=\frac{\log N}{N} $$ , characterised by a sharp peak in the specific heat capacity (SHC), which separates a high temperature from a low temperature region. The peak grows and the low-temperature region shrinks to zero with increasing N . In the micro-canonical ensemble, for finite N , there is a low energy phase with negative SHC and a high energy phase with positive SHC. The low-energy phase is dominated by a super-exponential growth of degeneracies as a function of energy which is directly related to the rapid growth in the number of directed graphs, with any number of vertices, as a function of the number of edges. The two ensembles have matching behaviour above the transition temperature. We further provide evidence that these thermodynamic properties hold in systems with U( N ) symmetry such as the zero charge sector of the 2-matrix model and in certain tensor models. We discuss the implications of these observations for the negative specific heat capacities in gravity using the AdS/CFT correspondence.- 20 December 2024
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2024 (12) , 1-67
Open AccessPermutation invariant matrix quantum thermodynamics and negative specific heat capacities in large N systems
A bstract: We study the thermodynamic properties of the simplest gauged permutation invariant matrix quantum mechanical system of oscillators, for general matrix size N . In the canonical ensemble, the model has a transition at a temperature T given by $$ x={e}^{-1/T}\sim {x}_c={e}^{-1/{T}_c}=\frac{\log N}{N} $$ , characterised by a sharp peak in the specific heat capacity (SHC), which separates a high temperature from a low temperature region. The peak grows and the low-temperature region shrinks to zero with increasing N . In the micro-canonical ensemble, for finite N , there is a low energy phase with negative SHC and a high energy phase with positive SHC. The low-energy phase is dominated by a super-exponential growth of degeneracies as a function of energy which is directly related to the rapid growth in the number of directed graphs, with any number of vertices, as a function of the number of edges. The two ensembles have matching behaviour above the transition temperature. We further provide evidence that these thermodynamic properties hold in systems with U( N ) symmetry such as the zero charge sector of the 2-matrix model and in certain tensor models. We discuss the implications of these observations for the negative specific heat capacities in gravity using the AdS/CFT correspondence. - Open Access
- 8 August 2024
- Published by Elsevier in Physica A: Statistical Mechanics and its Applications
- Vol. 651
Permutation invariant Gaussian matrix models for financial correlation matrices
- 8 August 2024
- Published by Elsevier in Physica A: Statistical Mechanics and its Applications
- Vol. 651
Open AccessPermutation invariant Gaussian matrix models for financial correlation matrices
- 6 August 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions
- Vol. 13 (3) , 579-623
Permutation invariant tensor models and partition algebras
Matrix models with continuous symmetry are powerful tools for studying quantum gravity and holography. Tensor models have also found applications in holographic quantum gravity. Matrix models with discrete permutation symmetry have been shown to satisfy large N factorisation properties relevant to holography, while also having applications to the statistical analysis of ensembles of real-world matrices. Here, we develop 3-index tensor models in dimension D with a discrete symmetry of permutations in the symmetric group S_{D} . We construct the most general permutation invariant Gaussian tensor model using the representation theory of symmetric groups and associated partition algebras. We define a representation basis for the 3-index tensors, where the two-point function is diagonalised. Inverting the change of basis gives an explicit formula for the two-point function in the tensor basis for general D .
- 6 August 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions
- Vol. 13 (3) , 579-623
Permutation invariant tensor models and partition algebras
Matrix models with continuous symmetry are powerful tools for studying quantum gravity and holography. Tensor models have also found applications in holographic quantum gravity. Matrix models with discrete permutation symmetry have been shown to satisfy large N factorisation properties relevant to holography, while also having applications to the statistical analysis of ensembles of real-world matrices. Here, we develop 3-index tensor models in dimension D with a discrete symmetry of permutations in the symmetric group S_{D} . We construct the most general permutation invariant Gaussian tensor model using the representation theory of symmetric groups and associated partition algebras. We define a representation basis for the 3-index tensors, where the two-point function is diagonalised. Inverting the change of basis gives an explicit formula for the two-point function in the tensor basis for general D .