Related Experiment Video
Updated: Sep 4, 2025

Fine-tuning the Size and Minimizing the Noise of Solid-state Nanopores
Published on: October 31, 2013
Noise correction of large deviations with anomalous scaling
Daniel Nickelsen1, Hugo Touchette2
1African Institute for Mathematical Sciences (AIMS), Muizenberg 7950, South Africa.
Abstract:
We present a path integral calculation of the probability distribution associated with the time-integrated moments of the Ornstein-Uhlenbeck process that includes the Gaussian prefactor in addition to the dominant path or instanton term obtained in the low-noise limit. The instanton term was obtained recently [D. Nickelsen and H. Touchette, Phys. Rev. Lett. 121, 090602 (2018)0031-900710.1103/PhysRevLett.121.090602] and shows that the large deviations of the time-integrated moments are anomalous in the sense that the logarithm of their distribution scales nonlinearly with the integration time. The Gaussian prefactor gives a correction to the low-noise approximation and leads us to define an instanton variance giving some insights as to how anomalous large deviations are created in time. The results are compared with simulations based on importance sampling, extending our previous results based on direct Monte Carlo simulations. We conclude by explaining why many of the standard analytical and numerical methods of large deviation theory fail in the case of anomalous large deviations.
Related Concept Videos
Testing a Claim about Standard Deviation
The hypothesis testing for the claim of population standard deviation (or variance) requires the data and samples to be random and unbiased. The population distribution also must be normal. There is no specific requirement on the sample size as the estimation is based on the chi-square distribution.
As a first step, the hypothesis (null and alternative) concerning the claim about...
Unsoundness of Aggregate due to Volume Change
Scaling
Regression Toward the Mean
Detection of Gross Error: The Q Test
NMR Spectrometers: Resolution and Error Correction

