Unitary matrix models and random partitions: Universality and multi-criticality

Kimura, Taro; Zahabi, Ali

doi:10.48550/arxiv.2105.00509

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…These asymptotic results have implications for the thermodynamics of quantum mechanical models based on tensor or multi-matrix models respectively. The multi-matrix case has been discussed in the context of AdS5/CFT4 and related gauge theories in [46] and more recently in [47][48][49][50]. The asymptotic counting we have done in this paper holds at large N. We are considering large n invariants when N ≫ n. The super-exponential growth of Z 3 (n) has the consequence of a vanishing Hagedorn temperature in this large n limit [10].…”

Section: Discussionmentioning

confidence: 98%

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Geloun

Ramgoolam

2022

J. Phys. A: Math. Theor.

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The counting of the dimension of the space of $U(N) \times U(N) \times U(N)$ polynomial invariants of a complex $3$-index tensor as a function of degree $n$ is known in terms of a sum of squares of Kronecker coefficients. For $n \le N$, the formula can be expressed in terms of a sum of symmetry factors of partitions of $n$ denoted $Z_3(n)$, which also counts the number of bipartite ribbon graphs with $n$ edges. We derive the large $n$ all-orders asymptotic formula for $ Z_3(n)$ making contact with high order results previously obtained numerically. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The derivation of the asymptotics relies on the dominance in the sum, of partitions with many parts of length $1$. This large $n$ dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general $d$-index tensors. The coefficients of powers of $ 1/n$ in these expansions involve Stirling numbers of the second kind along with restricted partition sums.

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Section: Discussionmentioning

confidence: 98%

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Geloun

Ramgoolam

2022

J. Phys. A: Math. Theor.

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show abstract

“…These asymptotic results have implications for the thermodynamics of quantum mechanical models based on tensor or multi-matrix models respectively. The multi-matrix case has been discussed in the context of AdS5/CFT4 and related gauge theories in [46] and more recently in [47,48,49,50]. The asymptotic counting we have done in this paper holds at large N. We are considering large n invariants when N ≫ n. The super-exponential growth of Z 3 (n) has the consequence of a vanishing Hagedorn temperature in this large n limit [10].…”

Section: Discussionmentioning

confidence: 98%

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Geloun¹,

Ramgoolam²

2021

Preprint

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The counting of the dimension of the space of U (N ) × U (N ) × U (N ) polynomial invariants of a complex 3-index tensor as a function of degree n is known in terms of a sum of squares of Kronecker coefficients. For n ≤ N , the formula can be expressed in terms of a sum of symmetry factors of partitions of n denoted Z 3 (n). We derive the large n all-orders asymptotic formula for Z 3 (n) making contact with high order results previously obtained numerically. The derivation relies on the dominance in the sum, of partitions with many parts of length 1. The dominance of other small parts in restricted partition sums leads to related asymptotic results. The result for the 3-index tensor observables gives the large n asymptotic expansion for the counting of bipartite ribbon graphs with n edges, and for the dimension of the associated Kronecker permutation centralizer algebra. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The large n dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general d-index tensors. The coefficients of 1/n in these expansions involve Stirling numbers of the second kind along with restricted partition sums.

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“…Recently, the unitary matrix models have attracted renewed interest in the context of matrix model/gauge theory correspondence [31,32,33,34,35,36]. (See also [37,38,39,40,40,41,42,43,44,45,46,47,48,49,50,51,52,53] for correspondence between more general matrix models and supersymmetric gauge theories.)…”

Section: Introductionmentioning

confidence: 99%

Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories

Oota

2021

Preprint

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Using the saddle point method, we give an explicit form of the planar free energy and Wilson loops of unitary matrix models in the one-cut regime. The multi-critical unitary matrix models are shown to undergo third-order phase transitions at two points by studying the planar free energy. One of these ungapped/gapped phase transitions is multi-critical, while the other is not multi-critical.The spectral curve of the k-th multi-critical matrix model exhibits an A 4k−1 singularity at the multi-critical point. Perturbation around the multi-critical point and its double scaling limit are studied. In order to take the double scaling limit, the perturbed coupling constants should be finetuned such that all the zero points of the spectral curve approach to the A 4k−1 singular point. The fine-tuning is examined in the one-cut regime, and the scaling behavior of the perturbed couplings is determined. It is shown that the double scaling limit of the spectral curve is isomorphic to the Seiberg-Witten curve of the Argyres-Douglas theory of type (A 1 , A 4k−1 ).

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Unitary matrix models and random partitions: Universality and multi-criticality

Cited by 3 publications

References 31 publications

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories

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