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Journal of Statistical Mechanics: Theory and Experiment Volume 2005 July 2005 Create an alert RSS this journal

Olivier Rivoire J. Stat. Mech. (2005) P07004 doi:10.1088/1742-5468/2005/07/P07004

The cavity method for large deviations

Olivier Rivoire

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Abstract References Cited By Figures

Figure 1

Figure 1. Rooted trees in the vertex-cover problem. This figure illustrates the recursion expressed by equation (19): on the left tree the root i (in the absence of 0) is constrained to be uncovered, while on the right tree it is constrained to be covered. The colouring in grey indicates that the nodes are neither constrained to be uncovered, nor to be covered.



Figure 2

Figure 2. Rate functions for the optimal energy epsilon in the vertex-cover problem with the binomial model \tilde {\G }^{(\gamma)}_N (label B), the Poissonian model \hat {\G }^{(\gamma)}_N (label P) and the uniform model \bar {\G }^{(\gamma)}_N (label U), all three for γ = 2. The common minimum at \bar {\e }\simeq 0.39 corresponds to the prediction of the typical RS cavity method (y = 0) [29]. The larger curvature of the rate function for the uniform model with respect to the two other models can be interpreted by the fact that fixing the ratio of edges imposes more constraints on the graph than fixing the degree distribution or edge probability.



Figure 3

Figure 3. Phase diagram of the vertex-cover problem in the (γ, y) plane for the Erdos-Rényi ensemble \tilde {\G }_N^{(\gamma)} . The full line corresponds to the line yc(γ) where the RS solution becomes instable; there is no instability below the percolation threshold, γ < 1 [y_c(\gamma)\to-\infty for \gamma \to 1^+ ]. The line y = 0 reproduces the phase diagram of the typical case with \gamma_c=e\simeq 2.71 being defined as the intersection of yc(γ) with y = 0.



Figure 4

Figure 4. Phase diagram of the vertex-cover problem in the (γ, epsilon) plane for the Erdos-Rényi ensemble \tilde {\G }_N^{(\gamma)} . The full line represents the typical RS energy density, as given by equation (27), and the thin line, starting only from γ = 1, the energy density above which the RS approach fails, as given by equation (31). The two curves cross at γc = e, corresponding to the onset of RSB on typical graphs.



Figure 5

Figure 5. Rate functions L(Σ0) of the complexity Σ0 in the three-colouring problem for the ensemble \tilde {\G }_N^{(\gamma)} with γ = 4.3 < γd, \gamma=4.45\in [\gamma_d,\gamma_c] and γ = 5 > γc, as obtained from the 1RSB-LDCM. A maximal value xd of the slope x=-\partial_{\Sigma_0} L(\Sigma_0) , associated with a maximal value of Σ0, is found above which a non-trivial L(Σ0) ceases to exist with, by definition of γd, xd(γ) < 0 when γ < γd, and xd(γ) > 0 when γ > γd. Note that for γ = 4.3, the curve displayed certainly does not describe the actual rate function which is expected to vanish at Σ0 = 0: since the non-trivial solution shown coexists with the trivial solution reduced to the point (Σ0 = 0, L = 0), the correct rate function may be obtained by a Maxwell construction, i.e., by drawing the supporting line originating from (Σ0 = 0, L = 0) and tangent to the curve.



Figure 6

Figure 6. Pictorial representation of the phase diagram for typical instances of the three-colouring problem on Erdos-Rényi graphs \tilde {\G }_N^{(\gamma)} . For γ < γd, the set of solutions forms a single connected cluster. For γd < γ < γc, the set of solutions is organized into exp(NΣ0) clusters, where Σ0 is the complexity curve represented in the upper part. For γ > γc, Σ0 < 0, indicating that there is (typically) no solution, i.e., the system is UNSAT.



Figure 7

Figure 7. Phase diagram of the three-colouring problem on Erdos-Rényi graphs in the (γ, x) plane. The top curve gives xd(γ), the value of x at which a 1RSB solution appears, and the bottom curve gives xc(γ), the value of x at which the complexity vanishes. The intersection of these two curves with the line x = 0 gives back the typical phase diagram of figure 6 with the intersection points γd and γc. (Note that this diagram includes values from non-physical branches as discussed in figure 5 for γ = 4.3.)




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