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On matrix realization of random surfaces

Technical Report:

Abstract

The large N one-matrix model with a potential, V({phi}) = {phi}{sup 2}/2 + g{sub 4}{phi}{sup 4}/N + g{sub 6}{phi}{sup 6}/N{sup 2} is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients r{sub k} (k = 1,2,3,...) of the orthogonal polynomials at large N. We find that for g{sub 6}/g{sub 4}{sup 2} > 1/2 there is no m = 2 solution which can be expressed as a smooth function of k/N in the limit N {yields} {infinity}. This means that the assumption of smoothness of r{sub k} at N {yields} {infinity} near the critical point, which was essential to derive the string susceptibility and the string equation, is broken even at the tree level of the genus expansion by adding the {phi}{sup 6}-term. We have also observed the free energy around the (expected) critical point to confirm that the system does not have the desired criticality as pure gravity. Our (discouraging) results for m = 2 are complementary to previous analyses by the saddle point method. On the other hand, for the case m = 3 (g{sub 6}/g{sub 4}{sup 2} = 4/5), we find a well-behaved solution which  More>>
Publication Date:
Dec 01, 1990
Product Type:
Technical Report
Report Number:
YITP/U-90-29
Reference Number:
SCA: 662130; 662110; PA: JPN-91:011017; SN: 92000630433
Resource Relation:
Other Information: PBD: Dec 1990
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; QUANTUM GRAVITY; MATRICES; STRING MODELS; TWO-DIMENSIONAL CALCULATIONS; REGGE CALCULUS; DEGREES OF FREEDOM; FEYNMAN DIAGRAM; SCALING LAWS; PARTITION FUNCTIONS; 662130; 662110; S-MATRIX THEORY, RELATIVISTIC SCATTERING THEORY; THEORY OF FIELDS AND STRINGS
OSTI ID:
10108960
Research Organizations:
Kyoto Univ., Uji (Japan). Yukawa Inst. for Theoretical Physics
Country of Origin:
Japan
Language:
English
Other Identifying Numbers:
Other: ON: DE92750826; TRN: JP9111017
Availability:
OSTI; NTIS (US Sales Only); INIS
Submitting Site:
JPN
Size:
35 p.
Announcement Date:
Jun 30, 2005

Technical Report:

Citation Formats

Sasaki, Misao, and Suzuki, Hiroshi. On matrix realization of random surfaces. Japan: N. p., 1990. Web.
Sasaki, Misao, & Suzuki, Hiroshi. On matrix realization of random surfaces. Japan.
Sasaki, Misao, and Suzuki, Hiroshi. 1990. "On matrix realization of random surfaces." Japan.
@misc{etde_10108960,
title = {On matrix realization of random surfaces}
author = {Sasaki, Misao, and Suzuki, Hiroshi}
abstractNote = {The large N one-matrix model with a potential, V({phi}) = {phi}{sup 2}/2 + g{sub 4}{phi}{sup 4}/N + g{sub 6}{phi}{sup 6}/N{sup 2} is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients r{sub k} (k = 1,2,3,...) of the orthogonal polynomials at large N. We find that for g{sub 6}/g{sub 4}{sup 2} > 1/2 there is no m = 2 solution which can be expressed as a smooth function of k/N in the limit N {yields} {infinity}. This means that the assumption of smoothness of r{sub k} at N {yields} {infinity} near the critical point, which was essential to derive the string susceptibility and the string equation, is broken even at the tree level of the genus expansion by adding the {phi}{sup 6}-term. We have also observed the free energy around the (expected) critical point to confirm that the system does not have the desired criticality as pure gravity. Our (discouraging) results for m = 2 are complementary to previous analyses by the saddle point method. On the other hand, for the case m = 3 (g{sub 6}/g{sub 4}{sup 2} = 4/5), we find a well-behaved solution which coincides with the result by Brezin et al.. To strengthen the validity of our numerical scheme, we present in appendix a non-perturbative solution for m = 1 which obeys the so-called type-II string equation proposed by Demeterfi et al.. (author).}
place = {Japan}
year = {1990}
month = {Dec}
}