Reiko Toriumi
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Reiko Toriumi has a total of 21 co-authors. Here are the top co-authors by number of shared publications:
- Nicolas Delporte
Okinawa Institute of Science and Technology Graduate University
5 shared publications - Giacomo La Scala
Okinawa Institute of Science and Technology Graduate University 1919-1 Tancha Onna Kunigami District 904-0495 Okinawa Japan
3 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
3 shared publications - Naoki Sasakura
Yukawa Institute for Theoretical Physics, Kyoto University , Kitashirakawa, Sakyo-ku, Kyoto 606-8502
3 shared publications - Saswato Sen
Okinawa Institute of Science and Technology Graduate University 1919-1 Tancha Onna Kunigami District 904-0495 Okinawa Japan
3 shared publications - Antonio D. Pereira
Universidade Federal Fluminense
2 shared publications
Latest Publications
- Open Access
- 10 June 2026
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2026 (6) , 119
Real eigenvalue/vector distributions of random real antisymmetric tensors
A bstract: Real eigenpairs of a real antisymmetric tensor of order p and dimension N can be defined as pairs of a real eigenvalue and p orthonormal N -dimensional real eigenvectors. We compute the signed and the genuine distributions of such eigenvalues of Gaussian random real antisymmetric tensors by using a quantum field theoretical method. An analytic expression for finite N is obtained for the signed distribution and the analytic large- N asymptotic forms for both. We compute the edge of the distribution for large- N , one application of which is to give an upper bound (believed tight) of the injective norm of the random real antisymmetric tensor. We find a large- N universality across various tensor eigenvalue distributions: the large- N asymptotic forms of the distributions of the eigenvalues z of the complex, complex symmetric, real symmetric, and real antisymmetric random tensors are all expressed by $$ {e}^{NB\;{h}_p\left({z}_c^2/{z}^2\right)+o(N)} $$ , where the function h p (·) depends only on the order p , while B and z c differ for each case, NB being the total dimension of the eigenvectors and z c being determined by the phase transition point of the quantum field theory.- 10 June 2026
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2026 (6) , 119
Open AccessReal eigenvalue/vector distributions of random real antisymmetric tensors
A bstract: Real eigenpairs of a real antisymmetric tensor of order p and dimension N can be defined as pairs of a real eigenvalue and p orthonormal N -dimensional real eigenvectors. We compute the signed and the genuine distributions of such eigenvalues of Gaussian random real antisymmetric tensors by using a quantum field theoretical method. An analytic expression for finite N is obtained for the signed distribution and the analytic large- N asymptotic forms for both. We compute the edge of the distribution for large- N , one application of which is to give an upper bound (believed tight) of the injective norm of the random real antisymmetric tensor. We find a large- N universality across various tensor eigenvalue distributions: the large- N asymptotic forms of the distributions of the eigenvalues z of the complex, complex symmetric, real symmetric, and real antisymmetric random tensors are all expressed by $$ {e}^{NB\;{h}_p\left({z}_c^2/{z}^2\right)+o(N)} $$ , where the function h p (·) depends only on the order p , while B and z c differ for each case, NB being the total dimension of the eigenvectors and z c being determined by the phase transition point of the quantum field theory. - Open Access
Critical Phenomena on the Bethe Lattice
We investigate the critical behavior of a family of $\mathbb{Z}_2$-symmetric scalar field theories on the Bethe lattice (the tree limit of regular hyperbolic tessellations) using both the non-perturbative Functional Renormalization Group and lattice perturbation theory. The family is indexed by the parameter $ζ\in (0,1]$, which determines the range of the theory via the kinetic term constructed from the graph Laplacian raised to the power $ζ$. Specifically, $ζ=1$ is the short-range theory, while $0<ζ<1$ defines the long-range model. Due to the hyperbolic nature of Bethe lattices, the Laplacian lacks a zero mode and exhibits a spectral gap. We find that upon closing this spectral gap by a modification of the Laplacian, the scalar field theories exhibit novel critical behavior in the form of non-trivial fixed points with critical exponents governed by $ζ$ and the spectral dimension $d_s=3$. In particular, our analysis indicates the presence of a Wilson-Fisher fixed point for the short range $ζ=1$ theory. In contrast, the nearest-neighbor Ising model on the Bethe lattice is known to exhibit mean-field critical exponents. To the best of our knowledge, this work provides the first evidence that a scalar $φ^4$ theory and the discrete Ising model on the same underlying lattice may lie in distinct universality classes.Open AccessCritical Phenomena on the Bethe Lattice
We investigate the critical behavior of a family of $\mathbb{Z}_2$-symmetric scalar field theories on the Bethe lattice (the tree limit of regular hyperbolic tessellations) using both the non-perturbative Functional Renormalization Group and lattice perturbation theory. The family is indexed by the parameter $ζ\in (0,1]$, which determines the range of the theory via the kinetic term constructed from the graph Laplacian raised to the power $ζ$. Specifically, $ζ=1$ is the short-range theory, while $0<ζ<1$ defines the long-range model. Due to the hyperbolic nature of Bethe lattices, the Laplacian lacks a zero mode and exhibits a spectral gap. We find that upon closing this spectral gap by a modification of the Laplacian, the scalar field theories exhibit novel critical behavior in the form of non-trivial fixed points with critical exponents governed by $ζ$ and the spectral dimension $d_s=3$. In particular, our analysis indicates the presence of a Wilson-Fisher fixed point for the short range $ζ=1$ theory. In contrast, the nearest-neighbor Ising model on the Bethe lattice is known to exhibit mean-field critical exponents. To the best of our knowledge, this work provides the first evidence that a scalar $φ^4$ theory and the discrete Ising model on the same underlying lattice may lie in distinct universality classes. - 12 November 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions
- Vol. 13 (2) , 281-355
Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^{2} \times \mathrm{O}(D)$ multi-matrix modelsThe authors studied in [Ann. Inst. Henri Poincaré D 9 (2022), 367–433], a complex multi-matrix model with \mathrm{U}(N)^{2} \times \mathrm{O}(D) symmetry, and whose double scaling limit where simultaneously the large- N and large- D limits were taken while keeping the ratio N/\sqrt{D}=M finite and fixed. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade \ell=0 was achieved. In this current study, we classify the higher order graphs in \ell . More specifically, \ell=1 and \ell=2 with arbitrary genus, in addition to a specific class of two-particle-irreducible (2PI) graphs for higher \ell \geqslant 3 but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single \mathrm{O}(D) -loop with an arbitrary \ell corresponds to a reduced alternating knot diagram with \ell crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I.
- 12 November 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions
- Vol. 13 (2) , 281-355
Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^{2} \times \mathrm{O}(D)$ multi-matrix modelsThe authors studied in [Ann. Inst. Henri Poincaré D 9 (2022), 367–433], a complex multi-matrix model with \mathrm{U}(N)^{2} \times \mathrm{O}(D) symmetry, and whose double scaling limit where simultaneously the large- N and large- D limits were taken while keeping the ratio N/\sqrt{D}=M finite and fixed. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade \ell=0 was achieved. In this current study, we classify the higher order graphs in \ell . More specifically, \ell=1 and \ell=2 with arbitrary genus, in addition to a specific class of two-particle-irreducible (2PI) graphs for higher \ell \geqslant 3 but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single \mathrm{O}(D) -loop with an arbitrary \ell corresponds to a reduced alternating knot diagram with \ell crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I.
- Open Access
Real eigenvalue/vector distributions of random real antisymmetric tensors
Real eigenpairs of a real antisymmetric tensor of order $p$ and dimension $N$ can be defined as pairs of a real eigenvalue and $p$ orthonormal $N$-dimensional real eigenvectors. We compute the signed and the genuine distributions of such eigenvalues of Gaussian random real antisymmetric tensors by using a quantum field theoretical method. An analytic expression for finite $N$ is obtained for the signed distribution and the analytic large-$N$ asymptotic forms for both. We compute the edge of the distribution for large-$N$, one application of which is to give an upper bound (believed tight) of the injective norm of the random real antisymmetric tensor. We find a large-$N$ universality across various tensor eigenvalue distributions: the large-$N$ asymptotic forms of the distributions of the eigenvalues $z$ of the complex, complex symmetric, real symmetric, and real antisymmetric random tensors are all expressed by $e^{N\,B\, h_p(z_c^2/z^2)+o(N)}$, where the function $h_p(\cdot)$ depends only on the order $p$, while $B$ and $z_c$ differ for each case, $NB$ being the total dimension of the eigenvectors and $z_c$ being determined by the phase transition point of the quantum field theory.Open AccessReal eigenvalue/vector distributions of random real antisymmetric tensors
Real eigenpairs of a real antisymmetric tensor of order $p$ and dimension $N$ can be defined as pairs of a real eigenvalue and $p$ orthonormal $N$-dimensional real eigenvectors. We compute the signed and the genuine distributions of such eigenvalues of Gaussian random real antisymmetric tensors by using a quantum field theoretical method. An analytic expression for finite $N$ is obtained for the signed distribution and the analytic large-$N$ asymptotic forms for both. We compute the edge of the distribution for large-$N$, one application of which is to give an upper bound (believed tight) of the injective norm of the random real antisymmetric tensor. We find a large-$N$ universality across various tensor eigenvalue distributions: the large-$N$ asymptotic forms of the distributions of the eigenvalues $z$ of the complex, complex symmetric, real symmetric, and real antisymmetric random tensors are all expressed by $e^{N\,B\, h_p(z_c^2/z^2)+o(N)}$, where the function $h_p(\cdot)$ depends only on the order $p$, while $B$ and $z_c$ differ for each case, $NB$ being the total dimension of the eigenvectors and $z_c$ being determined by the phase transition point of the quantum field theory. - Open Access
Characteristic polynomials of tensors via Grassmann integrals and distributions of roots for random Gaussian tensors
We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not totally symmetric ones. Drawing an analogy with matrix eigenvalues obtained from the roots of their characteristic polynomials, we study the roots of our tensor characteristic polynomial. Unlike standard definitions of eigenvalues of tensors of dimension $N$ giving $\sim e^{\text{constant} \, N}$ number of eigenvalues, our polynomial always has $N$ roots. For random Gaussian tensors, the density of roots follows a generalized Wigner semi-circle law based on the Fuss-Catalan distribution, introduced previously by Gurau [arXiv:2004.02660 [math-ph]].Open AccessCharacteristic polynomials of tensors via Grassmann integrals and distributions of roots for random Gaussian tensors
We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not totally symmetric ones. Drawing an analogy with matrix eigenvalues obtained from the roots of their characteristic polynomials, we study the roots of our tensor characteristic polynomial. Unlike standard definitions of eigenvalues of tensors of dimension $N$ giving $\sim e^{\text{constant} \, N}$ number of eigenvalues, our polynomial always has $N$ roots. For random Gaussian tensors, the density of roots follows a generalized Wigner semi-circle law based on the Fuss-Catalan distribution, introduced previously by Gurau [arXiv:2004.02660 [math-ph]]. - Open Access
- 21 August 2024
- Published by Springer Nature in The European Physical Journal C
- Vol. 84 (8) , 1-27
Counting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are constructed by contractions of complex tensors of order $$r+q$$ , also denoted (r, q). These tensors transform under r fundamental representations of the unitary group U(N) and q fundamental representations of the orthogonal group O(N). Therefore, $$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are tensor model observables endowed with a tensor field of order (r, q). We enumerate these observables using group theoretic formulae, for tensor fields of arbitrary order (r, q). Inspecting lower-order cases reveals that, at order (1, 1), the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order (r, q), the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $$r>1$$ , the number of invariants matches the number of (q-dependent) weighted equivalence classes of branched covers of the 2-sphere with r branched points. At $$r=1$$ , the counting maps to the enumeration of branched covers of the 2-sphere with $$q+3$$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.- 21 August 2024
- Published by Springer Nature in The European Physical Journal C
- Vol. 84 (8) , 1-27
Open AccessCounting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are constructed by contractions of complex tensors of order $$r+q$$ , also denoted (r, q). These tensors transform under r fundamental representations of the unitary group U(N) and q fundamental representations of the orthogonal group O(N). Therefore, $$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ invariants are tensor model observables endowed with a tensor field of order (r, q). We enumerate these observables using group theoretic formulae, for tensor fields of arbitrary order (r, q). Inspecting lower-order cases reveals that, at order (1, 1), the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order (r, q), the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $$r>1$$ , the number of invariants matches the number of (q-dependent) weighted equivalence classes of branched covers of the 2-sphere with r branched points. At $$r=1$$ , the counting maps to the enumeration of branched covers of the 2-sphere with $$q+3$$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments. - 20 June 2024
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 57 (27) , 275002
Dirac walks on regular trees
The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al (2022 Linear Multilinear Algebr. 72 325–65) proposed a classical ‘Dirac walk’ diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such ‘Dirac walks’ on the Bethe lattice, a d-regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ( d s = 1 for d = 2 and d s = 3 for d ⩾ 3 ).
- 20 June 2024
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 57 (27) , 275002
Dirac walks on regular trees
The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al (2022 Linear Multilinear Algebr. 72 325–65) proposed a classical ‘Dirac walk’ diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such ‘Dirac walks’ on the Bethe lattice, a d-regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ( d s = 1 for d = 2 and d s = 3 for d ⩾ 3 ).
- Open Access
- 7 May 2024
- Published by Springer Nature in Annales Henri Poincare
- Vol. 26 (3) , 947-1008
Dually Weighted Multi-matrix Models as a Path to Causal Gravity-Matter Systems
We introduce a dually-weighted multi-matrix model that for a suitable choice of weights reproduce two-dimensional Causal Dynamical Triangulations (CDT) coupled to the Ising model. When Ising degrees of freedom are removed, this model corresponds to the CDT-matrix model introduced by Benedetti and Henson (Phys Lett B 678:222, 2009). We present exact as well as approximate results for the Gaussian averages of characters of a Hermitian matrix A and $$A^2$$ for a given representation and establish the present limitations that prevent us to solve the model analytically. This sets the stage for the formulation of more sophisticated matter models coupled to two-dimensional CDT as dually weighted multi-matrix models providing a complementary view to the standard simplicial formulation of CDT-matter models.- 7 May 2024
- Published by Springer Nature in Annales Henri Poincare
- Vol. 26 (3) , 947-1008
Open AccessDually Weighted Multi-matrix Models as a Path to Causal Gravity-Matter Systems
We introduce a dually-weighted multi-matrix model that for a suitable choice of weights reproduce two-dimensional Causal Dynamical Triangulations (CDT) coupled to the Ising model. When Ising degrees of freedom are removed, this model corresponds to the CDT-matrix model introduced by Benedetti and Henson (Phys Lett B 678:222, 2009). We present exact as well as approximate results for the Gaussian averages of characters of a Hermitian matrix A and $$A^2$$ for a given representation and establish the present limitations that prevent us to solve the model analytically. This sets the stage for the formulation of more sophisticated matter models coupled to two-dimensional CDT as dually weighted multi-matrix models providing a complementary view to the standard simplicial formulation of CDT-matter models. - Open AccessCounting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are constructed by contractions of complex tensors of order $r+q$, also denoted $(r,q)$. These tensors transform under $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Therefore, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are tensor model observables endowed with a tensor field of order $(r,q)$. We enumerate these observables using group theoretic formulae, for arbitrary tensor fields of order $(r,q)$. Inspecting lower-order cases reveals that, at order $(1,1)$, the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order $(r,q)$, the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $r>1$, the number of invariants matches the number of ($q$-dependent) weighted equivalence classes of branched covers of the 2-sphere with $r$ branched points. At $r=1$, the counting maps to the enumeration of branched covers of the 2-sphere with $q+3$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.Open AccessCounting $U(N)^{\otimes r}\otimes O(N)^{\otimes q}$ invariants and tensor model observables$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are constructed by contractions of complex tensors of order $r+q$, also denoted $(r,q)$. These tensors transform under $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Therefore, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are tensor model observables endowed with a tensor field of order $(r,q)$. We enumerate these observables using group theoretic formulae, for arbitrary tensor fields of order $(r,q)$. Inspecting lower-order cases reveals that, at order $(1,1)$, the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order $(r,q)$, the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $r>1$, the number of invariants matches the number of ($q$-dependent) weighted equivalence classes of branched covers of the 2-sphere with $r$ branched points. At $r=1$, the counting maps to the enumeration of branched covers of the 2-sphere with $q+3$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.
- Open Access
Dually weighted multi-matrix models as a path to causal gravity-matter systems
We introduce a dually-weighted multi-matrix model that for a suitable choice of weights reproduce two-dimensional Causal Dynamical Triangulations (CDT) coupled to the Ising model. When Ising degrees of freedom are removed, this model corresponds to the CDT-matrix model introduced by Benedetti and Henson [Phys. Lett. B 678, 222 (2009)]. We present exact as well as approximate results for the Gaussian averages of characters of a Hermitian matrix $A$ and $A^2$ for a given representation and establish the present limitations that prevent us to solve the model analytically. This sets the stage for the formulation of more sophisticated matter models coupled to two-dimensional CDT as dually weighted multi-matrix models providing a complementary view to the standard simplicial formulation of CDT-matter models.Open AccessDually weighted multi-matrix models as a path to causal gravity-matter systems
We introduce a dually-weighted multi-matrix model that for a suitable choice of weights reproduce two-dimensional Causal Dynamical Triangulations (CDT) coupled to the Ising model. When Ising degrees of freedom are removed, this model corresponds to the CDT-matrix model introduced by Benedetti and Henson [Phys. Lett. B 678, 222 (2009)]. We present exact as well as approximate results for the Gaussian averages of characters of a Hermitian matrix $A$ and $A^2$ for a given representation and establish the present limitations that prevent us to solve the model analytically. This sets the stage for the formulation of more sophisticated matter models coupled to two-dimensional CDT as dually weighted multi-matrix models providing a complementary view to the standard simplicial formulation of CDT-matter models.