Figure 1

Exact cluster enumeration for a $3\times 3$ grid, with coupling $J=1.778$ (units of $k_{B}T$), corresponding to the critical value for an infinite grid [22]. (a) Cluster entropies $\Delta S(\Gamma )$ vs length $L(\Gamma )$ of the interaction path for perfect sampling; representative $\Gamma$ are shown for $L=2,4$ (labels refer to the grid). (b) Error ${ϵ}_S$ vs $\Theta$ for perfect sampling ($B=\infty$, full curve), and two random samples with $B={10}^7$ (dashed) and $B=4500$ (dotted curve) configurations. The accuracy on ${ϵ}_S$ and each $\Delta S$ is $\sim {10}^{-15}$.Reuse & Permissions

Figure 2

Histograms of $\Delta S(i_1,i_2,i_3)$ for the 1D-Ising ($J=4$, $h=-5$, units of $k_{B}T$) and independent spin (IS) models, for $N=50$ and three values of $B$. The distributions collapse onto each other after rescaling by the standard deviation of the IS cluster entropies, $\Delta S_{\mathrm{IS}}(B)\simeq [3(2p-1{)}^2/2p(1-p)](3/B{)}^{5/2}+O(1/B^3)$. Universality at small $\Delta S$ holds for larger cluster sizes ($=3$ here). Large-$\Delta S$ tails are not universal (not shown), and are specific to the interaction network; see Fig. 1a.Reuse & Permissions

Figure 3

(a) Fraction of nonzero interactions recognized by our procedure (squares) and by the PL algorithm (circles, from 10) vs intensity $J$ of couplings on a $7\times 7$ grid with 30% dilution; similar performances are obtained for $20\times 20$ grids. The critical coupling is $J_c\simeq 2.8$ (units of $k_{B}T$) [22]. (b) Error on the inferred couplings vs $\Theta$ for $M\times M$ grids at “criticality” ($J_c=1.778$, no dilution) and $B=4500$; the dependence on $M$ is reduced with periodic boundary conditions (not shown). Inset: ${ϵ}_c$ vs $\Theta$ for $M=7$. (c) Same as (b) for the Viana-Bray spin glass model [15] (connectivity 5 and random $J_{ij}$, uniform in $[-J^0;J^0]$; $J^0=4$ is larger than the spin glass critical coupling, $J_c^0\simeq 3.5$ [15]). Inset: ${ϵ}_c$ vs $\Theta$ for $N=50$.Reuse & Permissions

Figure 4

Performance of the inference procedure as a function of the threshold $\Theta$. (a) Errors ${ϵ}_p$ and ${ϵ}_c$, (b) number (bottom, left scale) and maximal size (top, right scale) of clusters, (c) entropy $S(\Theta )$. (d) Reconstructed vs experimental $p_i$ and $c_{ij}$ for ${\Theta }^{*}$ (error bars are calculated from $\chi$). Spin values are ${\sigma }_i=1$ if cell $i$ is active in a 20 ms-time bin, 0 otherwise [2]. Note that, while the correlations $c_{ij}$ are small, the ratios $c_{ij}/(p_i{p}_j)$ are of the order of unity, and so are the inferred couplings $J_{ij}$.Reuse & Permissions