q^{\beta\gamma_j}\quad . \end{equation} If ultrametricity holds, then the first inequality requires that $q^{\alpha\gamma_i}=q^{\alpha\beta}$, while the second inequality requires that $q^{\beta\gamma_j}=q^{\alpha\beta}$. Thus $q^{\alpha\gamma_i}=q^{\beta\gamma_j}$. But because $\alpha$, $\beta$, $\gamma_i$, and $\gamma_j$ are chosen randomly and independently, the two variables $q^{\alpha\gamma_i}=q^{\beta\gamma_j}$ are also independent. Because each of these is chosen from a {\it continuous\/} distribution $P$, the probability that the two overlap values can be identical is zero, and we arrive at a contradiction. The only way to avoid the contradiction is if the two strict inequalities in Eq.~(\ref{eq:ultra}) {\it cannot\/} occur simultaneously. This means that either $q^{\alpha\gamma_k}\le q^{\beta\gamma_k}$ for {\it every\/} $k$ or vice-versa, which implies that the pure states can be ordered into a one-dimensional continuum, and the ultrametric structure resembles a comb rather than a more usual tree, such as appears in the SK picture. As discussed in the previous section (see also the next section), self-averaging makes it implausible that the set of overlaps is countable. A countable set of overlaps would invalidate the above argument and possibly rescue ultrametricity, but at the cost of destroying anything resembling the Parisi solution. \medskip {\it Decomposition into pure states.\/} --- What is the nature of the decomposition of $\rho_{\cal J}$ into pure states? The SK picture prediction of a countably infinite sum as in Eq.~(\ref{eq:sum}) (with infinitely many $q_{\cal J}^{\alpha\beta}$'s) has been largely ruled out since the set of $q_{\cal J}^{\alpha\beta}$'s would be self-averaging, which seems unreasonable. Even if one were unwilling to rule out countability on that ground, there are other arguments, not presented here, which make that possibility even more unlikely. These arguments suggest that all $\rho_{\cal J}^{\alpha}$'s appearing in a countable decomposition would have the same even spin correlations. This certainly seems inconsistent with the expected presence of domain walls between pure states unrelated by a global spin flip. We conclude that in any reasonable scenario for $\rho_{\cal J}$, there should be at most one pair of pure states (related by a global spin flip) with {\it strictly\/} positive weight. In other words, either (a) $\rho_{\cal J}$ is pure, or (b) it is a sum of two pure states related by a global flip, or (c) it is an integral over pure states with none having strictly positive weight, or (d) it has one ``special'' pair of pure states with strictly positive weight and all the rest with zero weight. Case (a) occurs if the system is in a paramagnetic phase, or any other in which the EA order parameter is zero. Case (b) would occur according to the Fisher-Huse droplet picture \cite{FH1}, but could also occur if there existed multiple pure states not appearing in $\rho_{\cal J}$ (``weak Fisher-Huse'') \cite{Note4}. Case (c) occurs if there are {\it uncountably\/} many pure states in the decomposition of $\rho_{\cal J}$, all with zero weight (``democratic multiplicity''). Case (d) (which we regard as unlikely) occurs when one pair of pure states partially dominates all others, but accounts for only part of the total weight (``dictatorial multiplicity''). What is the relation between the nature of $P(=P_{\cal J})$ and the three (nontrivial) cases (b) -- (d) discussed above? Clearly case (b) implies that $P$ is a sum of two $\delta$-functions at $\pm q_{EA}$, and no continuous part. If we assume in cases (c) and (d) that varying $\alpha$ and $\beta$ through the continuous portion of the pure states yields a continuously varying $q_{\alpha\beta}$ (but see the next paragraph for a case where this assumption is violated), then it follows that case (c) corresponds to a $P$ with {\it no\/} $\delta$-functions while case (d) corresponds to a $P$ with $\delta$-functions at $\pm q_{EA}$ {\it and\/} a continuous part. This latter case is the $P$ predicted by the Parisi solution, but note two crucial distinctions between case (d) and the SK picture: (i) there is self-averaging, so one already obtains the continuous part of $P$ from a single realization ${\cal J}$, and (ii) the $\delta$-functions at $\pm q_{EA}$ come from a {\it single\/} special pair of pure states --- not from countably many $q^{\alpha\alpha}$'s. {\it Thus any numerical evidence in favor of such a distribution for\/} $\left[P_{\cal J}\right]_{\rm av}$ (as in Refs.~\cite{Caracciolo,Reger}) {\it is not evidence in favor of the SK picture\/} but rather, at most, supports dictatorial multiplicity. We remark that a case of democratic multiplicity occurs in a solution for the ground state structure in a short-ranged, highly disordered spin glass model \cite{NS94}. We argued there that below eight dimensions, there exists a single pair of ground states (case (b) above), while above eight, there are uncountably many. It is not hard to see that the $\rho_{\cal J}$ for $d>8$ corresponds to case (c) above -- the states are chosen by the flips of fair coins for all the trees in the invasion forest, so all have equal (zero) weight. It appears that here $P(q)$ is a $\delta$-function at zero! So even for short-ranged spin glasses with $T>0$, we cannot rule out the possibility of democratic multiplicity with such a $P$. \medskip {\it Discussion and Conclusions.} --- We have shown that in realistic spin glass models, and probably all non-infinite-ranged spin glasses, a natural construction leads to a Parisi overlap distribution $P_{\cal J}$ which is translation-invariant and hence self-averaging, unlike the Parisi solution of the SK model. Although our construction uses periodic boundary conditions, we believe that, in short-ranged (and probably all non-infinite-ranged) models, any choice of {\it coupling-independent\/} boundary conditions (e.g., free) should yield the same translation-invariant $P_{\cal J}$. Any restoration of the SK picture would require a non-translation-invariant $P_{\cal J}$ and we see no natural mechanism for obtaining one. Even were such a mechanism available, we note that the construction of our self-averaged $P_{\cal J}$ shows that the SK picture can be, at best, incomplete. Any theory of the thermodynamics of realistic spin glasses will likely differ considerably from the SK picture. If many phases do exist in some dimension and below some temperature, we believe that the most reasonable possibility is (c) above, i.e., democratic multiplicity. If numerical experiments find an order parameter distribution that looks similar to that of the Parisi solution, our arguments show that it should be interpreted within the context of possibility (d) above, i.e., dictatorial multiplicity, rather than as confirmation of the SK picture. To summarize our results, we have ruled out non-self-averaging in an extremely large class of disordered systems, which include short-ranged and probably all non-infinite-ranged spin glasses. Non-self-averaging, and the other main consequences of the Parisi solution, including ultrametricity of pure state overlaps, appear to be confined to mean field models. %\end{itemize} \small \begin{thebibliography}{10} \bibitem{BY} K.~Binder and A.P.~Young, \newblock {\em Rev.~Mod.~Phys.\/} {\bf 58}, 801 (1986). \bibitem{Bouchaud92} J.P.~Bouchaud, \newblock {\em J.~Phys.~I (France)} {\bf 2}, 1705 (1992). \bibitem{Bouchaud94} J.P.~Bouchaud, E.~Vincent, and J.~Hammann, \newblock {\em J.~Phys.~I (France)} {\bf 4}, 139 (1994). \bibitem {BM} A.J.~Bray and M.A.~Moore, \newblock in {\sl Heidelberg Colloquium in Glassy Dynamics}, ed. J.L.~van~Hemmen and I.~Morgenstern (Berlin: Springer-Verlag, 1987); A.J.~Bray and M.A.~Moore, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 58}, 57 (1987). \bibitem {Caracciolo} S.~Caracciolo, G.~Parisi, S.~Patarnello, and N.~Sourlas, \newblock {\em J.~Phys.~France\/} {\bf 51}, 1877 (1990). \bibitem {EA} S.~Edwards and P.W.~Anderson, \newblock {\em J.~Phys.~F\/} {\bf 5}, 965 (1975). \bibitem {FH1} D.S.~Fisher and D.A.~Huse, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 56}, 1601 (1986); D.S.~Fisher and D.A.~Huse, \newblock {\em Phys.~Rev.~B\/} {\bf 38}, 386 (1988). \bibitem{Franz} S.~Franz and M.~M\'ezard, \newblock {\em Physica A\/} {\bf 210}, 48 (1994). \bibitem {Georges} A.~Georges, M.~M\'ezard, and J.S.~Yedidia, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 64}, 2937 (1990). \bibitem {HF} D.A.~Huse and D.S.~Fisher, \newblock {\em J.~Phys.~A\/} {\bf 20}, L997 (1987); D.S.~Fisher and D.A.~Huse, \newblock {\em J.~Phys.~A\/} {\bf 20}, L1005 (1987). \bibitem {Lederman} M.~Lederman, R.~Orbach, J.M.~Hamann, M.~Ocio, and E.~Vincent, \newblock {\em Phys.~Rev.~B\/} {\bf 44}, 7403 (1991). \bibitem {Mac} W.L.~McMillan, \newblock {\em J.~Phys.~C\/} {\bf 17}, 3179 (1984). \bibitem {Mezard84} M.~M\'ezard, G.~Parisi, N.~Sourlas, G.~Toulouse, and M.~Virasoro, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 52}, 1156 (1984); {\em J.~Phys.~(Paris)\/} {\bf 45}, 843 (1984); A.P.~Young, A.J.~Bray, and A.M.~Moore, {\em J.~Phys.~C\/} {\bf 17}, L149 (1984); {\bf 17}, L155 (1984); M.~M\'ezard, G.~Parisi, and M.~Virasoro, {\em J.~Phys.~(Paris) Lett.\/} {\bf 46}, L217 (1985); B.~Derrida and G.~Toulouse, {\em J.~Phys.~(Paris) Lett.\/} {\bf 46}, L223 (1985). \bibitem{MPV} M.~M\'ezard, G.~Parisi, and M.A.~Virasoro, {\em Spin Glass Theory and Beyond\/} (World Scientific, Singapore, 1987.) \bibitem {NS92} C.M.~Newman and D.L.~Stein, \newblock {\em Phys.~Rev.~B} {\bf 46}, 973 (1992). \bibitem {NS94} C.M.~Newman and D.L.~Stein, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 72}, 2286 (1994). %\bibitem{NS3} C.M.~Newman and D.L.~Stein, %\newblock {\em Ann.~Inst.~Henri Poincar\'e\/} {\bf 31}, 249 (1995). %\bibitem{NS95} We emphasize that the discussion here refers only %to overlaps among pure states. In a recent paper, we show how %a tree-like structure can arise from very basic considerations for %the dynamics of systems with many {\it metastable\/} states. %See, for example, D.L.~Stein and C.M.~Newman, %\newblock{\em Phys.~Rev.~E\/} {\bf 51}, 5228 (1995). \bibitem {Parisi1} G.~Parisi, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 43}, 1754 (1979). \bibitem{Parisi2} G.~Parisi, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 50}, 1946 (1983); A.~Houghton, S.~Jain, and A.P.~Young, \newblock {\em J.~Phys.~C\/} {\bf 16}, L375 (1983). \bibitem{Parisi3} G.~Parisi, \newblock{\em Physica A\/} {\bf 194}, 28 (1993). \bibitem {Reger} J.D.~Reger, R.N.~Bhatt, and A.P.~Young, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 64}, 1859 (1990). \bibitem {SK} D.~Sherrington and S.~Kirkpatrick, \newblock {\em Phys.~Rev.~Lett.\/} {\bf 35}, 1972 (1975). \bibitem{Note1} For such models, however, there is no general theory of infinite volume states or their decompositions into pure phases. \bibitem {Note2} By translation-ergodicity, we simply mean that the spatial average of a random variable equals its expectation. This is equivalent to all translation-invariant random variables being constants. \bibitem{Note3} We caution that this should not be confused with ultrametric hopping dynamics between {\it metastable\/} states. For a discussion of how the latter can arise, see D.L.~Stein and C.M.~Newman, {\em Phys.~Rev.~E\/} {\bf 51}, 5228 (1995). \bibitem{Note4} This would be analogous to ``weak uniqueness'' for spin glasses, as discussed in M.~Campanino, E.~Olivieri, and A.C.D.~van Enter, {\em Comm.~Math.~Phys.\/} {\bf 108}, 241 (1987). \end{thebibliography} \end{document}