The Asymmetric Ground

The ground-state energy of a spin glass — the lowest-energy configuration in a disordered magnetic system — fluctuates from sample to sample. How unlikely is it for the ground-state energy to be significantly higher than typical?

The answer depends on whether an external magnetic field is present, and the dependence is exact: the rate function for large deviations above the typical ground-state energy is asymptotically quadratic near its minimum if and only if a magnetic field is applied.

Without a field, the spin glass has a continuous symmetry (flip all spins, energy unchanged). The large-deviation rate function has a non-quadratic shape near the minimum — the probability of atypical ground states decays in a pattern that doesn’t follow the Gaussian template. With a field, the symmetry breaks, and the rate function becomes quadratic — the standard Gaussian shape.

The proof uses a Parisi-type formula for fractional moments, extending the celebrated Parisi formula (which characterizes the typical free energy) to the large-deviation regime. The fractional moment controls how fast the probability of rare energy values decays, and its structure changes qualitatively when the field is turned on.

The through-claim: the presence of an external field doesn’t just shift the ground-state energy. It changes the geometry of the fluctuation landscape around the ground state. Quadratic rate functions mean that rare events are “normally distributed” in the large-deviation sense — the system deviates from typical in a smooth, predictable way. Non-quadratic rate functions mean the rare events have their own internal structure that the Gaussian approximation misses. The magnetic field doesn’t just break symmetry. It simplifies the statistics of the exceptional.