### Abstract

The large-N one-matrix model with a potential V()=22+g44N+g66N2 is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients rk (k=1, 2, 3, ) of the orthogonal polynomials at large N. We find that for g6g42>12 there is no m=2 solution which can be expressed as a smooth function of kN in the limit N. This means that the assumption of smoothness of rk at N 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 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 (g6g42=45), we find a well-behaved solution which coincides with the result obtained by Brézin, Marinari, and Parisi. To strengthen the validity of our numerical scheme, we present in an appendix a nonperturbative solution for m=1 which obeys the so-called type-II string equation.

Original language | English |
---|---|

Pages (from-to) | 4015-4028 |

Number of pages | 14 |

Journal | Physical Review D |

Volume | 43 |

Issue number | 12 |

DOIs | |

Publication status | Published - Jan 1 1991 |

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### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)

### Cite this

*Physical Review D*,

*43*(12), 4015-4028. https://doi.org/10.1103/PhysRevD.43.4015

**Matrix realization of random surfaces.** / Sasaki, Misao; Suzuki, Hiroshi.

Research output: Contribution to journal › Article

*Physical Review D*, vol. 43, no. 12, pp. 4015-4028. https://doi.org/10.1103/PhysRevD.43.4015

}

TY - JOUR

T1 - Matrix realization of random surfaces

AU - Sasaki, Misao

AU - Suzuki, Hiroshi

PY - 1991/1/1

Y1 - 1991/1/1

N2 - The large-N one-matrix model with a potential V()=22+g44N+g66N2 is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients rk (k=1, 2, 3, ) of the orthogonal polynomials at large N. We find that for g6g42>12 there is no m=2 solution which can be expressed as a smooth function of kN in the limit N. This means that the assumption of smoothness of rk at N 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 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 (g6g42=45), we find a well-behaved solution which coincides with the result obtained by Brézin, Marinari, and Parisi. To strengthen the validity of our numerical scheme, we present in an appendix a nonperturbative solution for m=1 which obeys the so-called type-II string equation.

AB - The large-N one-matrix model with a potential V()=22+g44N+g66N2 is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients rk (k=1, 2, 3, ) of the orthogonal polynomials at large N. We find that for g6g42>12 there is no m=2 solution which can be expressed as a smooth function of kN in the limit N. This means that the assumption of smoothness of rk at N 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 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 (g6g42=45), we find a well-behaved solution which coincides with the result obtained by Brézin, Marinari, and Parisi. To strengthen the validity of our numerical scheme, we present in an appendix a nonperturbative solution for m=1 which obeys the so-called type-II string equation.

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U2 - 10.1103/PhysRevD.43.4015

DO - 10.1103/PhysRevD.43.4015

M3 - Article

VL - 43

SP - 4015

EP - 4028

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 12

ER -