Large-deviation tails of critical order-parameter distributions

Technical Summary: Large-deviation tails of critical order-parameter distributions

Problem and Motivation
At continuous phase transitions, fluctuations are organized across all length scales. While standard finite-size scaling (FSS) effectively describes typical fluctuations via critical exponents, it does not fully characterize the rare, atypical macroscopic fluctuations found in the tails of critical probability distributions. These tails, described by large-deviation theory, reveal how the critical fixed point penalizes coherent order-parameter fluctuations beyond the typical scaling regime. Identifying the universal versus model-dependent features of these tails is an active area of research in critical phenomena. This work investigates the large-deviation tails of critical order-parameter distributions for percolation and Fortuin–Kasteleyn (FK) Ising models across two-dimensional (2D) and three-dimensional (3D) lattices, as well as on complete graphs (mean-field limit).

Methodology
The authors employ extensive Monte Carlo simulations to study two rescaled order parameters:

1. Magnetization-like variable ($x_m$): Defined as $x_m = |M|/\langle|M|\rangle$. For the FK-Ising model, $M$ is the standard spin sum. For percolation, where no microscopic spin exists, the authors construct a "signed cluster-mass" analogue: each cluster $a$ is assigned a random sign $S_a = \pm 1$, and $M = \sum S_a C_a$.
2. Largest-cluster variable ($x_C$): Defined as $x_C = C_1/\langle C_1 \rangle$, where $C_1$ is the size of the largest cluster. This applies to both percolation and the FK representation of the Ising model.

Simulations were conducted on:

• 2D square lattices ($L = 256, 512, 1024$).
• 3D simple cubic lattices ($L = 32, 64, 128$).
• Complete graphs with $N = 2^{16}, 2^{18}, 2^{20}$ sites.

For complete graphs, an accelerated bond-placement algorithm was used to handle the sparse connectivity at criticality efficiently. At least $10^7$ samples were generated for each configuration to ensure sufficient statistics in the rapidly decaying large-deviation tails.

Key Contributions and Theoretical Framework
The paper proposes and tests specific scaling forms for the tails of these distributions:

1. Magnetization Tail ($x_m$): The authors test the stretched-exponential form predicted by Eq. (3): $\tilde{P}(x) \sim \exp(-c x^{\delta+1})$, where $\delta+1 = 1/(1-y^*_h)$ and $y^*_h = y_h/d$. This form is expected to apply to both Ising and percolation-type variables.
2. Largest-Cluster Tail ($x_C$): Guided by exact results for complete graphs and scaling arguments, the authors propose a two-tail scaling form for the cumulative distribution function (CDF), $F_C(x)$:
   • Left tail ($x \to 0$): $F_C(x) \sim b_0 \exp(-b_1 x^{-1/y^*_h})$. This arises from the probability that all independent patches of the system remain below their typical cluster scale.
   • Right tail ($x \to \infty$): $1 - F_C(x) \sim c_0 x^{-1/y^*_h} \exp(-c_1 x^{1/(1-y^*_h)})$. This combines the algebraic prefactor from complete-graph results with the exponential cost governed by the magnetization scaling exponent.

Results

• Magnetization-like Variable ($x_m$): Numerical data for both 2D and 3D lattices, as well as complete graphs, support the stretched-exponential tail form. The 2D and 3D distributions exhibit the expected double-peak structure in the bulk, while the complete-graph limit shows a single peak. The FK-Ising model follows the mean-field quartic limit ($P(x) \propto \exp(-cx^4)$) on the complete graph, while the percolation pseudo-magnetization follows the corresponding mean-field tail ($\exp(-cx^3)$) with corrections in the central region due to the $Z_2$ symmetry of the signed construction.
• Largest-Cluster Variable ($x_C$): The proposed scaling forms for both the left and right tails of the CDF are validated across all geometries. The data collapse when plotted against the predicted powers of $x$ determined by the universality class-specific $y^*_h$.
• Complete-Graph FK-Ising Anomaly: A distinct finding concerns the left tail of the FK-Ising model on the complete graph. While the typical largest cluster scales as $N^{3/4}$ (Ising mean-field scaling), the far left tail is governed by rare configurations where the largest cluster scales as $N^{2/3}$ (percolation-like scaling). These rare events have a probability weight of order $N^{-1/12}$. The authors demonstrate that rescaling the distribution by $N^{1/12}$ and using the percolation scaling variable $x_p = C_1/N^{2/3}$ reveals a data collapse, confirming that the left tail probes a crossover to percolation-like behavior rather than typical Ising fluctuations.

Significance and Claims
The paper claims that the tails of order-parameter distributions reveal universal features of critical fluctuations that are not captured by averaged observables (such as the Binder ratio) alone. Specifically:

• The large-deviation tails provide a sensitive probe of universality classes, distinguishing between typical and rare fluctuation regimes.
• The signed cluster-mass variable successfully extends magnetization-like analysis to percolation, confirming that the same stretched-exponential tail structure applies to both spin and geometric models.
• The largest-cluster distribution serves as a geometric realization of the same large-deviation principles governing magnetization, with exponents fixed by the magnetic scaling dimension.
• The identification of the percolation-like left tail in the FK-Ising model on complete graphs highlights that rare configurations can follow scaling laws distinct from the typical phase behavior, offering a deeper understanding of the critical fixed point's structure.

The authors conclude that while the leading scaling forms are established, a more complete analytical theory including subleading prefactors and finite-size corrections remains an open direction for future work.