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The truncated Korteweg-De Vries (TKdV) system, a shallow-water wave modelwith Hamiltonian structure that exhibits weakly turbulent dynamics, has beenfound to accurately predict the anomalous wave statistics observed in recentlaboratory experiments. Majda et al. (2019) developed a TKdV statisticalmechanics framework based on a mixed Gibbs measure that is supported on asurface of fixed energy (microcanonical) and takes the usual macroconical formin the Hamiltonian. This paper reports two rigorous results regarding thesurface-displacement distributions implied by this ensemble, both in the limitof the cutoff wavenumber $\Lambda$ growing large. First, we prove that if theinverse temperature vanishes, microstate statistics converge to Gaussian as$\Lambda \to \infty$. Second, we prove that if nonlinearity is absent and theinverse-temperature satisfies a certain physically-motivated scaling law, thenmicrostate statistics converge to Gaussian as $\Lambda \to \infty$. When thescaling law is not satisfied, simple numerical examples demonstrate symmetric,yet highly non-Gaussian, displacement statistics to emerge in the linearsystem, illustrating that nonlinearity is not a strict requirement fornon-normality in the fixed-energy ensemble. The new results, taken together,imply necessary conditions for the anomalous wave statistics observed inprevious numerical studies. In particular, non-vanishing inverse temperatureand either the presence of nonlinearity or the violation of the scaling law arerequired for displacement statistics to deviate from Gaussian. The proof ofthis second theorem involves the construction of an approximating measure,which we find also elucidates the peculiar spectral decay observed in numericalstudies and may open the door for improved sampling algorithms.

Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics