Sample-to-sample fluctuations of the overlap distributions in the three-dimensional Edwards-Anderson spin glass

We study the sample-to-sample fluctuations of the overlap probability densities from large-scale equilibrium simulations of the three-dimensional Edwards-Anderson spin glass below the critical temperature. Ultrametricity, stochastic stability, and overlap equivalence impose constraints on the moments of the overlap probability densities that can be tested against numerical data. We found small deviations from the Ghirlanda-Guerra predictions, which get smaller as system size increases. We also focus on the shape of the overlap distribution, comparing the numerical data to a mean-field-like prediction in which finite-size effects are taken into account by substituting delta functions with broad peaks.

Figure 1

The quantity $X_{\mathrm{T}}$ as defined in the text, as a function of $q$ for lattice size $L=24$ (top) and $L=32$ (bottom) at temperature $T\simeq 0.64T_{\mathrm{c}}$. Insets show a magnified view of the region $q\sim 0.6$ (log-log plot). Plots show data for $X_{\mathrm{T}}$ computed only with triplets of independent configurations (ABC), with triplets in which two configurations belong to the same Monte Carlo history (AAB), and triplets in which all configurations come from the same Monte Carlo history (AAA). No significant difference shows up as long as we take enough uncorrelated configurations from the same replica.

Figure 2

Top: $X_2$ as a function of the corresponding polynomial in $X_1$ [Eq. (18)]. The straight line is the theoretical prediction (unit slope). Center: the ratio $X_2/X_1$ as a function of $X_1$, where the straight line is the theoretical prediction. Bottom: the squared difference $K_2={\left[X_2-(X_1+2X_1^2)/3\right]}^2$ as a function of $X_1$. Data refer to $T\sim 0.64T_{\mathrm{c}}$.

Figure 3

Data at $T\sim 0.64T_{\mathrm{c}}$. Top: $X_3$ as a function of the corresponding polynomial in $X_1$ and $X_{\mathrm{T}}$ [Eq. (19)]. The straight line is the theoretical prediction (unit slope). Bottom: the squared difference $K_3={[X_3-(2X_{\mathrm{T}}+2X_1+6X_1^2+5X_1^3)/15]}^2$ as a function of $X_1$, $T=0.64T_{\mathrm{c}}$. Lines connecting points are only a guide to the eye.

Figure 4

Top: the squared difference ${(X_{\mathrm{T}}-X_1^2)}^2$ as a function of $X_1$. Bottom: the quantity $K_3^u={[X_3-(2X_1+8X_1^2+5X_1^3)/15]}^2$ as a function of $X_1$. All data are for $T\sim 0.64T_{\mathrm{c}}$ and for lattice sizes $L=16,24,\mathrm{and}32$. The lines connecting the data points are only intended as a guide to the eye.

Figure 5

Square difference ${(X_{\mathrm{T}}-X_1^2)}^2$ (left) and the quantity $K_3^u={[X_3-(2X_1+8X_1^2+5X_1^3)/15]}^2$ (right) as a function of $X_1$. Top: for $T=0.75T_{\mathrm{c}}$ and $L=8,16,24,\mathrm{and}32$. Bottom: for $T=0.57T_{\mathrm{c}}$ and $L=8,16,\mathrm{and}24$.

Figure 6

Top: the conditioned probability $P\left(q_{12}\right|q_{34})$ (open squares) for $L=32$ and $T\sim 0.64T_{\mathrm{c}}$ and two values of $q_{34}=0.211$ (left) and $q_{34}=0.367$ (right). We also plot $2P\left(q_{12}\right)/3$ (open circles) and the difference (full triangles) of the two above quantities [Eq. (27) in the text], scaled by a factor of 2 for a better view. $q_{34}$ and $q_{\mathrm{EA}}$ values are indicated by vertical lines for visual reference. We took the value $q_{\mathrm{EA}}(L=32,T=0.64T_{\mathrm{c}})\sim 0.72$ as given in Ref.14. Bottom: the difference $P\left(q_{12}\right|q_{34})-2P\left(q_{12}\right)/3$ with $q_{34}=0.367$, for different lattice sizes compared at temperatures $T=0.75T_{\mathrm{c}}$, $0.64T_{\mathrm{c}}$, and $0.57T_{\mathrm{c}}$.

Figure 7

Asymptotic behavior of the cumulative probability ${\Pi }_q^C\left(z\right)$ [Eq. (23)]. Top: small-$z$ decay for $L=32$, $T=0.64T_{\mathrm{c}}$, and $q=0.3125$. Bottom: comparison of the exponent $x\left(q\right)$ obtained by the two methods described in the text [uppermost data points represent values obtained by fitting ${\Pi }_q^C\left(z\right)$, and lowermost data points come from integrating the $P\left(q\right)$], for some lattice sizes, many cutoff values $q$, and temperatures $T\sim 0.57T_{\mathrm{c}}$ (left) and $T\sim 0.64T_{\mathrm{c}}$ (right).

Figure 8

Comparison between the Monte Carlo data of the $P\left(q\right)$ and the convolution computed as described in the text (solid lines). Top: $L=32$, $T\sim 0.64T_{\mathrm{c}}$, and $T\sim 0.75T_{\mathrm{c}}$. Center: $L=24$, $T\sim 0.57T_{\mathrm{c}}$, and $T\sim 0.64T_{\mathrm{c}}$. Bottom: the conditioned probability $P\left(q_{12}\right|q_{34}=q_0)$ for $L=32$, $T\sim 0.64T_{\mathrm{c}}$, and some values of $q_0$.