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Center for Nonlinear Studies |
While macroscopic properties of hard-computational problems have been thoroughly investigated their manifestation in the corresponding microscopic configurations is much less understood. We introduce the notion of self-sustained clusters in Ising models, defined by the property that for each in-cluster spin, in-cluster field contribution dominates over the field induced by out-cluster spins; giving rise to highly stable configurations with respect to fluctuations. Self-sustained clusters shed light on the microscopic manifestation of metastable states and the emergence of frozen variables in constraint satisfaction problems. We discuss results obtained for the fully connected Ising and Sherrington-Kirkpatick (SK) models and present new results obtained for the p-spin model with p=3, which corresponds to the densely connected 3-XORSAT problem in the zero temperature limit; where the energy landscape undergoes an ergodicity breaking transition at a higher temperature than the spin-glass transition point. The analysis employs the (double) replica method to compute the entropy of macroscopic self-sustained clusters as a function of the temperature, their sizes and the difference between in-cluster and out-cluster induced fields. We observe that the self-sustained clusters are statically absent in the paramagnetic phase, but emerge at low temperatures when the computational problem becomes hard.
Host: Andrey Lokhov