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We determine the site and bond percolation thresholds for a system of

disordered jammed sphere packings in the maximally random jammed state,

generated by the Torquatoâ€“Jiao algorithm. For the site threshold, which

gives the fraction of conducting versus non-conducting spheres necessary

for percolation, we find \(p_c = 0.3116(3)\), consistent with the 1979 value of

Powell 0.310(5) and identical within errors to the threshold for the simplecubic

lattice, 0.311 608, which shares the same average coordination number

of 6. In terms of the volume fraction \(\phi \), the threshold corresponds to a critical

value \(\phi_c = 0.199\). For the bond threshold, which apparently was not measured

before, we find \(p_c = 0.2424(3)\). To find these thresholds, we considered two

shape-dependent universal ratios involving the size of the largest cluster,

fluctuations in that size, and the second moment of the size distribution; we

confirmed the ratiosâ€™ universality by also studying the simple-cubic lattice

with a similar cubic boundary. The results are applicable to many problems

including conductivity in random mixtures, glass formation, and drug loading

in pharmaceutical tablets.