Ground state of the Bethe lattice spin glass and running time of an exact optimization algorithm

We study the Ising spin glass on random graphs with fixed connectivity z and with a Gaussian distribution of the couplings, with mean $\ensuremath{\mu}$ and unit variance. We compute exact ground states by using a sophisticated branch-and-cut method for $z=4,6$ and system sizes up to 1280 spins, for different values of $\ensuremath{\mu}.$ We locate the spin-glass/ferromagnet phase transition at $\ensuremath{\mu}=0.77\ifmmode\pm\else\textpm\fi{}0.02(z=4)$ and $\ensuremath{\mu}=0.56\ifmmode\pm\else\textpm\fi{}0.02(z=6).$ We also compute the energy and magnetization in the Bethe-Peierls approximation with a stochastic method, and estimate the magnitude of replica symmetry breaking corrections. Near the phase transition, we observe a sharp change of the median running time of our implementation of the algorithm, consistent with a change from a polynomial dependence on the system size, deep in the ferromagnetic phase, to slower than polynomial in the spin-glass phase.

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