Geometric expansion of fluctuations and average shadows

TL;DR

The authors develop a systematic, geometry-driven method to determine the asymptotics of cumulants of local observables in large regions for translation-invariant systems. By introducing the volume covariogram and averaging over rotations, they derive explicit leading coefficients c0, c1, and c2 in the expansion C_m(λA)= c0 λ^d + c1 λ^{d-1} + c2 λ^{d-2} + ..., which factorize into geometric (intrinsic volumes of the convex hull of displacement vectors) and physical (convex moments of connected correlators) pieces. In two dimensions, c2 reduces to 2π χ_A times a convex moment, making odd cumulants for conserved quantities topological and independent of smooth deformations of A; this topological character is validated numerically in a 2D quantum Hall Laughlin state through Monte Carlo calculations of the third cumulant. The work unifies a broad class of cumulant expansions across smooth geometries and paves the way for higher-order terms and extensions to non-isotropic or curved spaces, with potential links to universal sum rules.

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.

Figures (6)

• Figure 1: (a) A large subregion $A$ in $\mathbb{R}^2$ with a smooth boundary $\partial A$. (b) Convex hull $H=H(\mathbf{r}_1,\ldots,\mathbf{r}_{m-1},0)$ relevant to the expansion of $C_m(A)$ in $d=2$ dimensions (an example with $m=7$ is shown). In this case, $2\pi v_1=\textrm{vol}(\partial H)$ is the perimeter of the hull, see \ref{['eq:v1_2d_3d']}, while $2\pi v_2=\textrm{vol}(H)$ is the area of the hull, see \ref{['eq:v2_2d_3d']}. (c) Mean shadow interpretation of $v_1$. Here $s(\theta)$ is the length of the shadow projected onto a fixed straight line for some orientation of $H$ labeled by the angle $\theta$ (the fixed light source is far away in a direction perpendicular to the line). Averaging $s(\theta)$ over all angles yields the mean width, which is twice $v_1$. (d) Convex hull $H$ relevant to the computation of $C_m(A)$ in $d=3$ dimensions (an example with $m=8$ is shown). In this case, $v_1$ is given by \ref{['eq:v1_2d_3d']} and involves the exterior dihedral angles between the two faces adjacent to the edges, while $8\pi v_2=\textrm{vol}(\partial H)$ (see \ref{['eq:v2_2d_3d']}) is the surface area of the hull, i.e. the sum of the surface area of all faces.
• Figure 2: Left: a corner with angle $\theta$ contributes a constant term $a_m(\theta)$ to all cumulants. Right: sequence of convex regular polygons $A_N$, with interior angles $\theta_N=\pi(N-2)/N$ and fixed perimeter, the $N\rightarrow\infty$ limit of which is a disk.
• Figure 3: Illustration of the volume $\mathcal{G}_A$ in two dimensions. Left: region $A$ with interior shown in light green, and smooth boundary. Center: volume $\mathcal{G}_A(\mathbf{r})=\textrm{vol}[A\cap(A-\mathbf{r})]$ shown in green, for the choice of vector $\mathbf{r}$ (red arrow). Right: volume $\mathcal{G}_A(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3)=\textrm{vol}[A\cap(A-\mathbf{r}_1)\cap (A-\mathbf{r}_2)\cap(A-\mathbf{r}_3)]$ in green, for the choice of vectors $\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3$ (red, blue, brown arrows).
• Figure 4: Illustration of the isotropic volume $\overline{\mathcal{G}_A}(\mathbf{r}_1,\mathbf{r}_2)$ in two dimensions. The choice of region $A$ is the same as in the previous figure, the vectors $\mathbf{r}_1,\mathbf{r}_2$ are shown top left in red and blue. The translated regions $A-\mathbf{r}_1$ and $A-\mathbf{r}_2$ are shown in lighter colors to better identify the initial region $A$. The volume $\mathcal{G}_A(\mathbf{r}_1,\mathbf{r}_2)$ is represented in green. From top left to top right and bottom right to bottom left: $\mathcal{G}_A(\mathbf{r}_1^\theta,\mathbf{r}_2^\theta)$ where the $\mathbf{r}_j^\theta$ are obtained from the $\mathbf{r}_j$ by a rotation of angle $\theta$, with successive clockwise increments by $\pi/4$ shown. $\overline{\mathcal{G}_A}(\mathbf{r}_1,\mathbf{r}_2)$ is obtained by averaging $\mathcal{G}_A(\mathbf{r}_1^\theta,\mathbf{r}_2^\theta)$ over all angles $\theta$. Vectors $\mathbf{r}_1$ and $\mathbf{r}_2$ were chosen to be not so small to magnify curvature effects.
• Figure 5: Exact same setup as in Fig. \ref{['fig:multivolume']}, but for $\mathcal{V}_A$ instead of $\mathcal{G}_A$. Left: region $A$. Middle: volume $\mathcal{V}_A(\mathbf{r})$ of $A\cap(B-\mathbf{r}_1)$ shown in green. Right: volume $\mathcal{V}_A(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3)$ of $A\cap[(B-\mathbf{r}_1)\cup (B-\mathbf{r}_2)\cap(B-\mathbf{r}_3)]$ (green) used to compute the fourth cumulant $C_4(A)$ in two dimensions. For small $\mathbf{r}_j$ most of the volume is localized near the boundary $\partial A$.