Stylized Facts and Their Microscopic Origins: Clustering, Persistence, and Stability in a 2D Ising Framework

TL;DR

The paper develops a two-dimensional Ising-like network of agents with Glauber dynamics, introducing a global mean-field term controlled by $\alpha$ and a microscopic stability factor (MSF) to connect micro-scale cluster structure to macro-scale price dynamics. By analyzing cluster morphology, persistence, and MSF across temperature-like control $1/\beta$, the work shows that near the critical point $1/\beta_c$ clusters occur over multiple scales with a power-law distribution (exponent ~$1.9$) and MSF exhibits intermediate memory, producing heavy-tailed returns and volatility clustering. In low-temperature regimes, large persistent clusters yield low volatility, while high-temperature regimes lead to rapid, uncorrelated reconfigurations and Gaussian-like returns; when global coupling is present ($\alpha=4$), the MSF and clusters display intermittent memory and abrupt reorganizations, echoing endogenous market crises. Overall, the study provides a microscopic mechanism whereby cluster dynamics and their memory produce stylized facts such as sharp returns, heavy tails, and zero autocorrelation, highlighting the role of internal structure in financial market behavior.

Abstract

The analysis of financial markets using models inspired by statistical physics offers a fruitful approach to understand collective and extreme phenomena [3, 14, 15] In this paper, we present a study based on a 2D Ising network model where each spin represents an agent that interacts only with its immediate neighbors plus a term reated to the mean field [1, 2]. From this simple formulation, we analyze the formation of spin clusters, their temporal persistence, and the morphological evolution of the system as a function of temperature [5, 19]. Furthermore, we introduce the study of the quantity $1/2P\sum_{i}|S_{i}(t)+S_{i}(t+Δt)|$, which measures the absolute overlap between consecutive configurations and quantifies the degree of instantaneous correlation between system states. The results show that both the morphology and persistence of the clusters and the dynamics of the absolute sum can explain universal statistical properties observed in financial markets, known as stylized facts [2, 12, 18]: sharp peaks in returns, distributions with heavy tails, and zero autocorrelation. The critical structure of clusters and their reorganization over time thus provide a microscopic mechanism that gives rise to the intermittency and clustered volatility observed in prices [2, 15].

Figures (18)

• Figure 1: Spin configurations obtained from the dynamics of the two-dimensional Ising model at $1/\beta=0.5$ and $\alpha=4$. Each configuration corresponds to a sample taken every 1000 time steps, with the instants 5k, 50k, 100k, and 290k being arbitrarily selected. At this $1/\beta$, the system exhibits a coexistence of large and small domains that evolve slowly over time. Although the general structure of the clusters is preserved, fluctuations and gradual shifts of the domain boundaries are observed, reflecting a slow relaxation dynamic toward more ordered states (Color online).
• Figure 2: Below $1/\beta$, the +1 clusters (red) exhibit a steeper slope in the distribution, indicating a predominance of small clusters, while the -1 clusters (blue) show a heavier distribution, with a higher probability of finding large clusters. The dispersion of the points suggests oscillations in cluster formation, reflecting the active dynamics of the domains at this temperature (Color online).
• Figure 3: The cluster size distribution at $1/\beta=2.2$ exhibits a power-law behavior, indicating the absence of a characteristic scale. In this regime, the dynamics are highly variable and allow for the simultaneous coexistence of clusters of all sizes, from the smallest to the largest (Color online).
• Figure 4: Close to the critical “ temperature” $1/\beta=T=2.2$ ($1/\beta_{c}=T_{c}=2.26$). A critical state is observed in the Ising model (or any system with competing domains). None of the phases dominates. The cluster size distribution follows a power law with an exponent of 1.9, typical of criticality (Color online).
• Figure 5: At high $1/\beta=10$, the clusters remain small and unstable. Intense thermal agitation causes them to constantly form and dissolve, generating configurations dominated by small, ephemeral clusters, as shown in the figure (Color online).