Pinning Susceptibility: The effect of dilute, quenched disorder on jamming

We study the effect of dilute pinning on the jamming transition. Pinning reduces the average contact number needed to jam unpinned particles and shifts the jamming threshold to lower densities, leading to a pinning susceptibility, $\chi_p$. Our main results are that this susceptibility obeys scaling form and diverges in the thermodynamic limit as $\chi_p \propto |\phi - \phi_c^\infty|^{-\gamma_p}$ where $\phi_c^\infty$ is the jamming threshold in the absence of pins. Finite-size scaling arguments yield these values with associated statistical (systematic) errors $\gamma_p = 1.018 \pm 0.026 (0.291)$ in $d=2$ and $\gamma_p =1.534 \pm 0.120 (0.822)$ in $d=3$. Logarithmic corrections raise the exponent in $d=2$ to close to the $d=3$ value, although the systematic errors are very large.