Equilibrium states of the spin glass on a Bethe lattice

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Abstract
The existence of many equilibrium states of the spin glass on a Bethe lattice at low temperatures is confirmed numerically, directly on the lattice. Following a method due to Nemoto and Takayama (1985), approximate solutions of the exact equations for the Bethe lattice site magnetisations (mi) are found by minimising the norm mod Del F mod identical to ( Sigma i( delta F/ delta mi)2)1/2 where F((mi)) is the free energy. In order to examine the stability of solutions on the Bethe lattice using the Hessian delta 2/F delta mi delta mj, it is necessary to connect up boundary sites. Evidence is presented for the bifurcation of equilibrium states with decreasing temperature.