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Contributed by: C.M. Newman (Courant Inst.) and D.L. Stein (Univ. Arizona), Sept. 18, 1998.

Consider a simple Edwards-Anderson Ising spin glass, which is an Ising model on Zd, whose nearest neighbor couplings are independent, identically distributed random variables with mean zero and finite variance (e.g., Gaussian). In the absence of an external magnetic field, the following are open questions (all statements below are understood to apply to almost every coupling realization):

1) Prove (or disprove) the widely held belief that the model has a thermodynamic phase transition above some lower critical dimension dc -in the sense that the infinite-volume Gibbs states at low temperature T differ, either in number or qualitative nature, from that at high T. If so, find dc.

2) Prove (or disprove) that if a transition exists, then for some d and T the low-temperature phase breaks spin flip symmetry, i.e., that all (or at least some) pure (i.e., extremal) states come in distinct pairs that map into each other under a global spin flip.

The next set of questions concerns the thermodynamics of the low-temperature phase, if such a distinct phase exists.

3) For a given d and T, find the number of pure state pairs.

4) If there are (infinitely) many, then it has been argued heuristically ([1]) that the spin glass metastate, defined as a probability measure on the Gibbs states ([2]) is dispersed over infinitely many mixed states each consisting of a single pair whose members have equal weights.

Roughly speaking, this means that in each large finite cube with, say, periodic boundary conditions, one only ``sees'' a single pair of pure states, but this pair changes with volume. In particular, more complicated mixtures of states (which would exhibit relative domain walls), such as those predicted in various mean-field pictures, would not appear in the metastate.

Prove (or disprove) this statement.


5) If multiple Gibbs states are shown to exist, determine whether the low-temperature phase persists in the presence of a small external magnetic field.

Further discussion by the authors, and other references, are found in:
  1. C.M. Newman and D.L. Stein, Phys. Rev. E 57 (1997) 1356 -- 1366.

  2. C.M. Newman and D.L. Stein, Phys. Rev. E 55 (1998) 5194-5211, and in
    ``Mathematical Aspects of Spin Glasses and Neural Networks'', A. Bovier and P. Picco, eds. (Birkh\"auser, Boston, 1998), pp. 243 -- 287.

  3. C. M. Newman, ``Topics in Disordered Systems'', Birkhauser, Basel, 1997.

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newman@cims.nyu.edu, dls@physics.arizona.edu (contributors),
aizenman@princeton.edu (editor)