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Geometric expansion of fluctuations and average shadows
by Clément Berthière, Benoit Estienne, Jean-Marie Stéphan, William Witczak-Krempa
Submission summary
| Authors (as registered SciPost users): | Clément Berthière · Jean-Marie Stéphan |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2408.08364v3 (pdf) |
| Date accepted: | Oct. 14, 2025 |
| Date submitted: | Oct. 8, 2025, 9:10 p.m. |
| Submitted by: | Clément Berthière |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Fluctuations of observables provide unique insights into the nature of physical systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this paper, we develop a systematic method to determine the asymptotic behavior of cumulants of local observables as the region becomes large. Our analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. A particular application of our method shows that, in two dimensions, the leading behavior of odd cumulants of conserved quantities is topological, specifically depending on the Euler characteristic of the region. We illustrate these results with the paradigmatic strongly-interacting system of two-dimensional quantum Hall state at filling fraction $1/2$, by performing Monte-Carlo calculations of the skewness (third cumulant) of particle number in the Laughlin state.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
As suggested by the referees, we did a refactoring of our paper (in particular section III) for a streamlined presentation and a better flow.
We now discuss how the generality of our results extend beyond gapped phases, as well as what we know of the physical content behind the convex moments.
List of changes
- Structure improved (Section III in particular)
- Discussion expanded
- The convention to compute volumes that was given below Eq. (29) is now given explicitly below Eq. (12)
Published as SciPost Phys. 19, 122 (2025)