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The low-temperature Sherrington–Kirkpatrick fluctuation law
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GAME #217
The low-temperature Sherrington–Kirkpatrick fluctuation law
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| The low-temperature Sherrington–Kirkpatrick fluctuation law. For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to $c_\beta n^{1/3}$, with $c_\beta\gt 0$, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law. |
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For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n1/3 converges to a finite positive constant.
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For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is $n^{1/6+o(1)}$. The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.
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