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The jamming transition of frictionless soft spheres is singular because it coincides with the threshold for mechanical stability. This lends the marginally-jammed solid just above the transition special properties, including excess low-frequency modes that extend all the way down to zero frequency at the transition. The low-frequency excess modes are quasi-localized and extremely sensitive to mechanical loading. They also have low energy barriers to rearrangements, suggesting that high-displacement regions in low-frequency quasi-localized modes may serve to identify shear transformation zones.
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