Saburo Higuchi

Renormalization group approach to multiple-arc random matrix models

Abstract We study critical and universal behaviors of unitary invariant non-gaussian random matrix ensembles within the framework of the large- N renormalization group. For a simple double-well model we find an unstable fixed point and a stable inverse-gaussian fixed point. The former is identified as the critical point of single/double-arc phase transition with a discontinuity of the third derivative of the free energy. The latter signifies a novel universality of large- N correlators other than the usual single arc type. This phase structure is consistent with the universality classification of two-level correlators for multiple-arc models by Ambjorn and Akemann. We also establish the stability of the gaussian fixed point in the multi-coupling model.

Feynman rules for string field theories with discrete target space

Abstract We derive a minimal set of Feynman rules for the loop amplitudes in unitary models of closed strings, whose target space is a simply laced (extended) Dynkin diagram. The string field Feynman graphs are composed of propagators, vertices (including tadpoles) of all topologies, and leg factors for the macroscopic loops. A vertex of given topology factorizes into a fusion coefficient for the matter fields and an intersection number associated with the corresponding punctured surface. As illustration we obtain explicit expressions for the genus-one tadpole and the genus-zero four-loop amplitude.

COMPACT POLYMERS ON DECORATED SQUARE LATTICES

A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It serves as a model of a compact polymer on a lattice. I study the number of Hamiltonian cycles, or equivalently the entropy of a compact polymer, on various lattices that are not homogeneous but with a sublattice structure. Estimates for the number are obtained by two methods. One is the saddle point approximation for a field theoretic representation. The other is the numerical diagonalization of the transfer matrix of a fully packed loop model in the zero fugacity limit. In the latter method, several scaling exponents are also obtained.

FIELD THEORETIC APPROACH TO THE COUNTING PROBLEM OF HAMILTONIAN CYCLES OF GRAPHS

A Hamiltonian cycle of a graph is a closed path that visits each site once and only once. I study a field theoretic representation for the number of Hamiltonian cycles for arbitrary graphs. By integrating out quadratic fluctuations around the saddle point, one obtains an estimate for the number which reflects characteristics of graphs well. The accuracy of the estimate is verified by applying it to 2d square lattices with various boundary conditions. This is the first example of extracting meaningful information from the quadratic approximation to the field theory representation.

One-dimensional double chain composed of carbamoylmethylthio-substituted TTF-based donor in ion radical salt

Carbamoylmethylthio-substituted TTF-based donors afforded inclusion-type salts with a BF4 counter ion or an F4-TCNQ acceptor. The crystal structures of these salts are characterized by a one-dimensional double chain composed of singly-oxidized donors via SOMO–SOMO interaction and side-by-side S⋯S contacts. Counter ions of BF4 are hydrogen-bonded to the amide groups. On the other hand, F4-TCNQ acceptors are stacked in a channel created by a stair-like arrangement of donor chains. The thermally activating magnetic susceptibility of the F4-TCNQ complex was interpreted in terms of a double-chain frustrated spin system.

COUNTING HAMILTONIAN CYCLES ON PLANAR RANDOM LATTICES

A Hamiltonian cycle of a graph is a closed path if it passs through each of the vertices once and only once. In this letter, Hamiltonian cycles on planar random lattices are considered. The generating function for the number of Hamiltonian cycles is obtained and its singularity is studied. Its relation with two-dimensional quantum gravity is discussed.

Loop model with generalized fugacity in three dimensions

A statistical model of loops on the three-dimensional lattice is proposed and is investigated. It is O(n )-type but has loop fugacity that depends on global three-dimensional shapes of loops in a particular fashion. It is shown that, despite this nonlocality and the dimensionality, a layer-to-layer transfer matrix can be constructed as a product of local vertex weights for infinitely many points in the parameter space. Using this transfer matrix, the site entropy is estimated numerically in the fully packed limit.

HAMILTONIAN CYCLES ON RANDOM LATTICES OF ARBITRARY GENUS

Abstract A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It has been difficult to count the number of Hamiltonian cycles on regular lattices with periodic boundary conditions, e.g. lattices on a torus, due to the presence of winding modes. In this paper, the exact number of Hamiltonian cycles on a random trivalent fat graph drawn faithfully on a torus is obtained. This result is further extended to the case of random graphs drawn on surfaces of an arbitrary genus. The conformational exponent y is found to depend on the genus linearly.

A CONNECTION BETWEEN LATTICE AND SURGERY CONSTRUCTIONS OF THREE-DIMENSIONAL TOPOLOGICAL FIELD THEORIES

We study the relation between lattice construction and surgery construction of three-dimensional topological field theories. We show that a class of the Chung-Fukuma-Shapere theory on the lattice has representation theoretic reformulation which is closely related to the Altschuler-Coste theory constructed by surgery. There is a similar relation between the Turaev-Viro theory and the Reshetikhin-Turaev theory.