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Fluctuations in Various Regimes of Non-Hermiticity and a Holographic Principle

G. Akemann, M. Duits, L. D. Molag

TL;DR

The paper analyzes fluctuations of non-interacting fermions trapped in low dimensions through determinantal point processes, focusing on the elliptic Ginibre ensemble that interpolates between GUE and Ginibre via the parameter $\tau$. It develops a holographic principle linking the entropy (Renyi and von Neumann) to the boundary length of a set and proves a central limit theorem that interpolates between Hermitian and non-Hermitian regimes on mesoscopic scales. A detailed steepest-descent analysis yields precise variance formulas for smooth linear statistics in both strong and weak non-Hermiticity limits, including edge/cusp corrections captured by the conformal map $\psi$ and an obstacle-function framework; a Ward-identity approach then establishes Gaussian fluctuations with a variance that decomposes into bulk and boundary contributions and remains valid in the transition regime $\alpha=\gamma$. Collectively, the results unify bulk, edge, and mesoscopic fluctuations across Hermitian and non-Hermitian limits and quantify how boundary geometry governs information-theoretic and counting statistics in random normal matrices.

Abstract

The variance of the number of particles in a set is an important quantity in understanding the statistics of non-interacting fermionic systems in low dimensions. An exact map of their ground state in a harmonic trap in one and two dimensions to the classical Gaussian unitary and complex Ginibre ensemble, respectively, allows to determine the counting statistics at finite and infinite system size. We will establish two new results in this setup. First, we uncover an interpolating central limit theorem between known results in one and two dimensions, for linear statistics of the elliptic Ginibre ensemble. We find an entire range of interpolating weak non-Hermiticity limits, given by a two-parameter family for the mesoscopic scaling regime. Second, we considerably generalize the proportionality between the number variance and the entanglement entropy between Fermions in a set $A$ and its complement in two dimensions. Previously known only for rotationally invariant sets and external potentials, we prove a holographic principle for general non-rotationally invariant sets and random normal matrices. It states that both number variance and entanglement entropy are proportional to the circumference of $A$.

Fluctuations in Various Regimes of Non-Hermiticity and a Holographic Principle

TL;DR

The paper analyzes fluctuations of non-interacting fermions trapped in low dimensions through determinantal point processes, focusing on the elliptic Ginibre ensemble that interpolates between GUE and Ginibre via the parameter . It develops a holographic principle linking the entropy (Renyi and von Neumann) to the boundary length of a set and proves a central limit theorem that interpolates between Hermitian and non-Hermitian regimes on mesoscopic scales. A detailed steepest-descent analysis yields precise variance formulas for smooth linear statistics in both strong and weak non-Hermiticity limits, including edge/cusp corrections captured by the conformal map and an obstacle-function framework; a Ward-identity approach then establishes Gaussian fluctuations with a variance that decomposes into bulk and boundary contributions and remains valid in the transition regime . Collectively, the results unify bulk, edge, and mesoscopic fluctuations across Hermitian and non-Hermitian limits and quantify how boundary geometry governs information-theoretic and counting statistics in random normal matrices.

Abstract

The variance of the number of particles in a set is an important quantity in understanding the statistics of non-interacting fermionic systems in low dimensions. An exact map of their ground state in a harmonic trap in one and two dimensions to the classical Gaussian unitary and complex Ginibre ensemble, respectively, allows to determine the counting statistics at finite and infinite system size. We will establish two new results in this setup. First, we uncover an interpolating central limit theorem between known results in one and two dimensions, for linear statistics of the elliptic Ginibre ensemble. We find an entire range of interpolating weak non-Hermiticity limits, given by a two-parameter family for the mesoscopic scaling regime. Second, we considerably generalize the proportionality between the number variance and the entanglement entropy between Fermions in a set and its complement in two dimensions. Previously known only for rotationally invariant sets and external potentials, we prove a holographic principle for general non-rotationally invariant sets and random normal matrices. It states that both number variance and entanglement entropy are proportional to the circumference of .

Paper Structure

This paper contains 26 sections, 46 theorems, 353 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup: Determinantal point processes, linear statistics and elliptic Ginibre ensemble
  3. The number variance and a holographic principle
  4. Smooth linear statistics of the elliptic Ginibre ensemble
  5. Mesoscopic fluctuations in various regimes of weak non-Hermiticity
  6. Plan of the paper
  7. The number variance of random normal matrices
  8. The number variance for constant $\Delta V$
  9. The number variance for general potentials
  10. The number variance at the edge of the elliptic Ginibre ensemble
  11. Estimating the entropy
  12. A steepest descent analysis for the elliptic Ginibre ensemble
  13. Summary of previous results: saddle points and other relevant expressions
  14. A single integral representation for the weak non-Hermiticity regime
  15. Smooth linear statistics in the strong non-Hermiticity regime
  16. ...and 11 more sections

Key Result

Theorem 1.1

Consider a random normal matrix model with a $C^2$ potential $V$ that satisfies eq:Vgrowth and which is assumed to be real analytic in a neighborhood of $S_V$. Fix a compact set $K\subset \mathring S_V$ and assume that $\Delta V>0$ on $K$. Then we have as $n\to\infty$ for all convex sets $A\subset K$ with a $C^2$ boundary, where the implied constants depend only on $V$ and $K$. When $\Delta V$ is

Figures (3)

  • Figure 1: The upper full line $\gamma=\frac{1+\alpha}{2}$ represents the microcopic scale near the origin, meaning that when $(1-\tau)\sim n^{-\alpha}$ we find that the microscopic scale is at $\sim n^{-(1+\alpha)/2}$. The mesoscopic scales are then at $\sim n^{-\gamma}$ with $0<\gamma<(1+\alpha)/2$. We obtain the variance for the mesosopic linear statistics of GUE type if $\alpha>\gamma$ in \ref{['eq:ThmI.6iii']} and of Ginibre (GinUE) type if $\alpha<\gamma$ in \ref{['eq:ThmI.6i']}. On the lower full line at $\alpha=\gamma$ we have a transition given in \ref{['eq:ThmI.6ii']} that depends on the weak non-Hermiticity parameter $\kappa=(1-\tau)n^\alpha$, and we prove a CLT.
  • Figure 2: Example of a set $A$ satisfying Assumption \ref{['asump:AphiPoints']}. The upper boundary curve is given by the graph of $\varphi_+$ while the lower boundary curve is given by the graph of $\varphi_-$, where $a_1\leq x\leq a_2$. Similarly, the right boundary curve is given by the graph of $\tilde{\varphi}_+$ while the left boundary curve is given by the graph of $\tilde{\varphi}_-$, where $b_1\leq y\leq b_2$.
  • Figure 3: On the left: The contour $\gamma_i$ for the single integral representation in Proposition \ref{['prop:alternativeSingleInt']}. The integrand has a jump on $(-i\infty,-i\sqrt 2]\cup [i\sqrt 2, i\infty)$. On the right: The contour (the deformation of $\gamma_i$) for the single integral representation \ref{['eq:IntRepTildeF']} suitable for the weak non-Hermiticity regime. The direction of the path close to $t=0$ is chosen such that it passes through $\tilde{a}$ and $-\tilde{a}$ in the allowed sectors. As $n\to\infty$ we have $\pm\tilde{a}\to 0$. Furthermore, the image of the contour from $-\infty$ to $\infty$ is invariant under a rotation by $\pi$ radians. The remaining contour is the steepest descent path going through the two saddle points $\tilde{b}_1, \tilde{b}_2$ in the upper half-plane that are not close to $0$.

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Holography
  • Theorem 1.4: Limiting variance for fixed $\tau$
  • Remark 1.5
  • Theorem 1.6: Limiting variance at weak non-Hermiticity
  • Theorem 1.7
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 82 more