## 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
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## 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}

}

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}

}