NDLI: Analyticity of the Planar Limit of a Matrix Model

Content Provider	Springer Nature Link
Author	Garoufalidis, Stavros Popescu, Ionel
Copyright Year	2012
Abstract	Using Chebyshev polynomials combined with some mild combinatorics, we provide an alternative approach to the analytical and formal planar limits of a random matrix model with a 1-cut potential V. For potentials $${V(x)=x^{2}/2-\sum_{n\ge1}a_{n}x^{n}/n}$$ , as a power series in all a n , the formal Taylor expansion of the analytic planar limit is exactly the formal planar limit. In the case V is analytic in infinitely many variables {a n } n ≥ 1 (on the appropriate spaces), the planar limit is also an analytic function in infinitely many variables and we give quantitative versions of where this is defined. Particularly useful in enumerative combinatorics are the gradings of $${V,V_{t}(x)=x^{2}/2-\sum_{n\ge1}a_{n}t^{n/2}x^{n}/n}$$ and $${V_{t}(x)=x^{2}/2-\sum_{n\ge3}a_{n}t^{n/2 -1}x^{n}/n}$$ . The associated planar limits F(t) as functions of t count planar diagram sorted by the number of edges respectively faces. We point out a method of computing the asymptotic of the coefficients of F(t) using the combination of the wzb method and the resolution of singularities. This is illustrated in several computations revolving around the important extreme potential $${V_{t}(x)=x^{2}/2+\log(1-\sqrt{t}x)}$$ and its variants. This particular example gives a quantitative and sharp answer to a conjecture of ’t Hooft’s, which states that if the potential is analytic, the planar limit is also analytic.
Ending Page	565
Page Count	67
Starting Page	499
File Format	PDF
ISSN	14240637
e-ISSN	14240661
Journal	Annales Henri Poincaré
Issue Number	3
Volume Number	14
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2012-07-31
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Topology of general $3$-manifolds Mathematical Methods in Physics Theoretical, Mathematical and Computational Physics Quantum Physics Dynamical Systems and Ergodic Theory Knots and links in $S^3$ Classical and Quantum Gravitation, Relativity Theory Elementary Particles, Quantum Field Theory
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Nuclear and High Energy Physics Mathematical Physics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Torus Knots and Mirror Symmetry

Nonrelativistic Hydrogen Type Stability Problems on Nonparabolic 3-Manifolds

Chern–Simons Invariants of Torus Links

Torus Knots in Lens Spaces and Topological Strings

Static Potentials on Asymptotically Flat Manifolds

Renormalizability Conditions for Almost-Commutative Manifolds

Van Hove Limit for Infinite Systems

A Limit Equation Criterion for Solving the Einstein Constraint Equations on Manifolds with Ends of Cylindrical Type

Diffusive Limit for a Quantum Linear Boltzmann Dynamics

Analyticity of the Planar Limit of a Matrix Model

Similar Documents

Torus Knots and Mirror Symmetry

Nonrelativistic Hydrogen Type Stability Problems on Nonparabolic 3-Manifolds

Chern–Simons Invariants of Torus Links

Torus Knots in Lens Spaces and Topological Strings

Static Potentials on Asymptotically Flat Manifolds

Renormalizability Conditions for Almost-Commutative Manifolds

Van Hove Limit for Infinite Systems

A Limit Equation Criterion for Solving the Einstein Constraint Equations on Manifolds with Ends of Cylindrical Type

Diffusive Limit for a Quantum Linear Boltzmann Dynamics

Analyticity of the Planar Limit of a Matrix Model