We consider the number distribution of topological defects resulting from the finite-time crossing of a continuous phase transition and identify signatures of universality beyond the mean value, predicted by the Kibble-Zurek mechanism. Statistics of defects follows a binomial distribution with N Bernouilli trials associated with the probability of forming a topological defect at the locations where multiple domains merge. All cumulants of the distribution are predicted to exhibit a common universal power-law scaling with the quench time in which the transition is crossed. Knowledge of the distribution is used to discussed the onset of adiabatic dynamics and bound rare events associated with large deviations.

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Paper published as part of the special topic on Excitons: Energetics and Spatio-temporal Dynamics This paper was selected as an Editor's Pick ARTICLES YOU MAY BE INTERESTED INEfficient evaluation of exact exchange for periodic systems via concentric atomic density fitting

A sequence of works in unconstrained online convex optimisation have investigated the possibility of adapting simultaneously to the norm U of the comparator and the maximum norm G of the gradients. In full generality, matching upper and lower bounds are known which show that this comes at the unavoidable cost of an additive GU 3 , which is not needed when either G or U is known in advance. Surprisingly, recent results by Kempka et al. (2019) show that no such price for adaptivity is needed in the specific case of 1-Lipschitz losses like the hinge loss. We follow up on this observation by showing that there is in fact never a price to pay for adaptivity if we specialise to any of the other common supervised online learning losses: our results cover log loss, (linear and non-parametric) logistic regression, square loss prediction, and (linear and non-parametric) least-squares regression. We also fill in several gaps in the literature by providing matching lower bounds with an explicit dependence on U . In all cases we obtain scale-free algorithms, which are suitably invariant under rescaling of the data. Our general goal is to establish achievable rates without concern for computational efficiency, but for linear logistic regression we also provide an adaptive method that is as efficient as the recent non-adaptive algorithm by Agarwal et al. (2021).