Rooks on ferrers boards and matrix integrals

1999 ◽  
Vol 96 (5) ◽  
pp. 3531-3536 ◽  
Author(s):  
S. V. Kerov
Keyword(s):  
Matrix Integrals ◽  
Ferrers Boards

scholarly journals The replica limit of unitary matrix integrals

2001 ◽  
Vol 592 (3) ◽  
pp. 419-444 ◽  
Author(s):  
D. Dalmazi ◽  
J.J.M. Verbaarschot
Keyword(s):  
Unitary Matrix ◽  
Matrix Integrals

scholarly journals Matrix integrals & finite holography

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Dionysios Anninos ◽  
Beatrix Mühlmann
Keyword(s):  
Critical Exponents ◽  
Continuum Theory ◽  
Point Of View ◽  
Leading Order ◽  
Minimal Models ◽  
Point Evaluation ◽  
Brst Cohomology ◽  
Matrix Integrals ◽  
The Matrix ◽  
The Continuum

Abstract We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

scholarly journals On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

10.37236/6699 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yu-Chang Liang ◽  
Tsai-Lien Wong
Keyword(s):  
Weyl Algebra ◽  
Stirling Numbers ◽  
Stirling Number ◽  
Combinatorial Structures ◽  
Normal Ordering ◽  
Rook Placements ◽  
Threshold Graphs ◽  
Combinatorial Interpretations ◽  
Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

scholarly journals UNITARY MATRIX MODELS AND 2D QUANTUM GRAVITY

1992 ◽  
Vol 07 (29) ◽  
pp. 2753-2762 ◽  
Author(s):  
S. DALLEY ◽  
C. V. JOHNSON ◽  
T. R. MORRIS ◽  
A. WÄTTERSTAM
Keyword(s):  
Matrix Models ◽  
Integrable Hierarchies ◽  
Renormalization Group Equation ◽  
Closed String ◽  
Singular Solutions ◽  
Unitary Matrix ◽  
Group Equation ◽  
Gravitational System ◽  
Large N ◽  
Matrix Integrals

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system — open-closed string theory. Non-perturbative solutions of the multicritical unitary matrix models map to non-singular solutions of the 'renormalization group' equation for the string susceptibility, [Formula: see text]. We also demonstrate that the large-N solutions of unitary matrix integrals in external fields, studied by Gross and Newman, equal the non-singular pure closed-string solutions of [Formula: see text].

Rook Theory. I.: Rook Equivalence of Ferrers Boards

1975 ◽  
Vol 52 (1) ◽  
pp. 485 ◽  
Author(s):  
Jay R. Goldman ◽  
J. T. Joichi ◽  
Dennis E. White
Keyword(s):  
Rook Theory ◽  
Ferrers Boards ◽  
Rook Equivalence

Unitary integrals and related matrix models

2018 ◽  
Author(s):  
Nicolas Orantin
Keyword(s):  
Matrix Models ◽  
Unitary Matrix ◽  
Technical Tool ◽  
Basic Properties ◽  
Matrix Integrals ◽  
Unitary Model ◽  
M Theory ◽  
Pair Correlator

This article examines the basic properties of unitary matrix integrals using three matrix models: the ordinary unitary model, the Brézin-Gross-Witten (BGW) model and the Harish-Chandra-Itzykson-Zuber (HCIZ) model. The tricky sides of the story are given special attention, such as the de Wit-’t Hooft anomaly in unitary integrals and the problem of correlators with Itzykson-Zuber measure. The method of character expansions is also emphasized as a technical tool. The article first provides an overview of the theory of the BGW model, taking into account the de Wit-’t Hooft anomaly and the M-theory of matrix models, before discussing the theory of the HCIZ integral. In particular, it describes the basics of character calculus, character expansion of the HCIZ integral, character expansion for the BGW model and Leutwyler-Smilga integral, and pair correlator in HCIZ theory.

Sign in / Sign up

Export Citation Format

Share Document