Critical Jammed Phase of the Linear Perceptron

Criticality in statistical physics naturally emerges at isolated points in the phase diagram. Jamming of spheres is not an exception: varying density, it is the critical point that separates the unjammed phase where spheres do not overlap and the jammed phase where they cannot be arranged without overlaps. The same remains true in more general constraint satisfaction problems with continuous variables where jamming coincides with the (protocol dependent) satisfiability transition point. In this work we show that by carefully choosing the cost function to be minimized, the region of criticality extends to occupy a whole region of the jammed phase. As a working example, we consider the spherical perceptron with a linear cost function in the unsatisfiable jammed phase and we perform numerical simulations which show critical power laws emerging in the configurations obtained minimizing the linear cost function. We develop a scaling theory to compute the emerging critical exponents.

Figure 1

The phase diagram of the spherical perceptron with linear cost function. The $y$ axis represents the density of constraints $\alpha =M/N$, while the $x$ axis is the control parameter $\sigma$ that defines the gap variables; see Eq. (1). The red solid line is the SAT-UNSAT (jamming) line (computed under the replica symmetric approximation, exact only for $\sigma >0$). Below this line the model is SAT, unjammed and one can find zero energy configurations. Above, the problem is UNSAT, jammed and the energy is positive. Above the black dashed line replica symmetry spontaneously breaks down: below this line the problem is convex, while above, the energy landscape is glassy with many local minima with critical properties. The jamming line lies in the RS region for $\sigma >0$ and in the RSB region for $\sigma <0$. Inset: The density of contacts $c$ for $\alpha =5$ as a function of $\sigma$. The red line corresponds to the theoretical prediction. We have hypostaticity $c<1$ in the RS phase and isostaticity, i.e., $c=1$, in the RSB phase. The dots come from numerical simulations with $N=1024$ at $\alpha =5$.

Figure 2

The cumulative distribution function (CDF) of both strictly positive and strictly negative gaps for $N=1024$ and $\alpha =5$. We compare the distributions in the UNSAT-RSB and UNSAT-RS regions. The RSB data refer to minima at an average energy $H/N=1.01\pm 0.02$, and the corresponding value of $\sigma$ is $\sigma =0.219\pm 0.004$. Both positive and negative gaps’ distributions are compatible with a power law at small argument with exponent $1-{\gamma }_J\simeq 0.59$ (black full line). The RS data refer to minima with $H/N=2.540\pm 0.013$ and $\sigma =0.757\pm 0.010$. One sees there a linear behavior of the CDF, implying a positive probability density function in the origin.

Figure 3

The CDF of the virtual forces presented as a function of both $\widehat{f}$ and $1-\widehat{f}$ for the same parameters as in Fig. 2. In the RSB phase the distribution of forces vanishes as a power law both in $\widehat{f}=0$ and in $\widehat{f}=1$ (purple line in the inset), and both powers are compatible with ${\theta }_J\simeq 0.42$ corresponding to a CDF with a power 1.42. In the RS region the behavior is linear and the PDF is finite at both edges of its support (yellow line in the inset). The datasets presented in the inset correspond to $⟨H⟩/N\simeq 2,3$ for RSB and RS curves respectively.