Eigenvalue Spectrum Phase Transition in Large-Cap US Equity Returns: Testing the Gravity-Model Analogy
Question
Does the eigenvalue spectrum of large-cap US equity returns exhibit a phase transition—a breakdown of spectral coherence—that marks the empirical boundary where the gravity-model analogy (treating market cap as mass and correlation distance as gravitational distance) ceases to predict structural equivalence and hedging relationships?
Method
We computed the eigenvalue spectrum of the return correlation matrix for 15 large-cap US equities (AAPL, AMZN, BAC, CVX, GOOGL, JNJ, JPM, KO, META, MSFT, NVDA, PEP, PFE, PG, WMT, XOM) over the window 2010-01-01 to 2024-12-31, using daily adjusted-close returns from yfinance (n = 3772 observations). The data source is yfinance daily adjusted-close returns for the named tickers over the stated window.
The method is principal component analysis (PCA) eigenvalue spectrum of the return correlation matrix versus the Marchenko-Pastur null distribution. Under the Marchenko-Pastur (MP) random-matrix theory, a correlation matrix constructed from uncorrelated Gaussian time series with q = n_assets / n_obs = 0.004 has eigenvalues bounded between a lower edge (0.8779) and an upper edge (1.1301). Eigenvalues above the MP upper bound are statistically distinguishable from random-matrix noise and represent genuine common factors. We counted the number of eigenvalues exceeding 1.1301 to determine the number of significant factors. To assess time variation, we recomputed the spectrum in-sample within each calendar year (2010–2024) on the same data and method, yielding a per-year significant-factor count.
This is an in-sample analysis; no out-of-sample validation or forward prediction is attempted. The computation tests whether the spectral structure is stable or exhibits a phase transition—a discrete jump in the number of significant factors—that would signal a breakdown in the gravity-model analogy's predictive power for structural equivalence.
Result
The full-period (2010–2024) eigenvalue spectrum contains three significant factors (eigenvalues above the MP upper bound of 1.1301). The top ten eigenvalues are 6.4878, 1.7134, 1.4928, 0.7632, 0.7174, 0.6574, 0.5596, 0.4912, 0.4377, and 0.3942. The first eigenvalue (6.4878) explains 43.25% of total variance; the three significant factors together explain 64.63% of variance.
The top factor (factor 1) loads most heavily on JPM (0.291), MSFT (0.290), and PEP (0.278). The second factor loads most heavily (in absolute value) on AMZN (−0.416), NVDA (−0.403), and GOOGL (−0.351). The negative sign indicates these assets load oppositely to the first factor's direction; economically, factor 2 captures a tech-growth versus diversified-value contrast, with high-beta technology names loading negatively.
The per-year significant-factor count reveals clear time variation:
- 2010–2012: 1 significant factor per year.
- 2013–2016: oscillation between 1 and 2 factors (2013: 2, 2014: 2, 2015: 1, 2016: 2).
- 2017: jump to 3 factors.
- 2018–2020: reversion to 2 factors.
- 2021–2024: stable at 3 factors.
The transition from 1–2 factors (2010–2016) to a stable 3-factor regime (2021–2024) constitutes an empirical phase transition in spectral coherence. The intermediate period (2017–2020) exhibits volatility in factor count, with 2017 and 2021 marking the onset and consolidation of the higher-dimensional regime.
Interpretation
The computed eigenvalue spectrum does exhibit a phase transition: a discrete, sustained increase in the number of significant factors from 1–2 (2010–2016) to 3 (2021–2024). This transition marks a breakdown in the low-dimensional structure that the gravity-model analogy presupposes. Under a strict gravity analogy, market cap (mass) and correlation distance (gravitational distance) would predict a single dominant factor (the market) plus noise, or at most a stable two-factor structure (market + sector). The emergence of a third persistent factor in 2021–2024 signals that correlation distance is no longer well-approximated by a simple mass-weighted gravitational potential.
The factor loadings clarify the economic content of the transition. Factor 1 is a broad market factor, loading positively on diversified names (JPM, MSFT, PEP). Factor 2 is a tech-growth versus value contrast, with AMZN, NVDA, and GOOGL loading negatively. The third factor (not detailed in loadings but implied by the eigenvalue count) likely captures a residual sectoral or style dimension that became statistically distinguishable from noise only after 2020. The gravity model, which treats all large-cap equities as point masses in a homogeneous financial space, cannot generate this third dimension without additional structure (e.g., sector-specific gravitational constants or non-Euclidean geometry).
The time variation in factor count is the key result. The gravity analogy predicts structural equivalence: assets with similar market caps should exhibit similar correlation distances to all other assets, and hedging relationships should be stable over time. The observed phase transition—particularly the jump from 2 factors (2020) to 3 factors (2021–2024)—contradicts this prediction. The 2021 transition coincides with the post-pandemic shift in monetary policy and the divergence of technology-sector valuations from broader market multiples, suggesting that the third factor captures a regime-dependent risk dimension that the gravity model's static mass-distance framework cannot accommodate.
The in-sample nature of the result is a limitation: we have not tested whether the three-factor structure predicts out-of-sample returns or hedging efficacy. The result establishes that spectral coherence broke down (factor count increased), but it does not establish that this breakdown degraded the gravity model's predictive power for future structural equivalence. That would require an out-of-sample test of hedging error or return prediction conditional on the factor structure.
The result does NOT support the claim that the gravity model is universally invalid. It supports the narrower claim that the model's predictive power for structural equivalence is time-varying and that a discrete phase transition occurred around 2021. The model may still hold approximately in the 1–2 factor regime (2010–2016) and fail in the 3-factor regime (2021–2024). The boundary is empirical, not theoretical.
Relation to the Literature
No closely related papers were retrieved, so the result stands on the computation alone. The gravity-model analogy in financial space is a conceptual framework, not a widely tested empirical hypothesis in the academic literature. The Marchenko-Pastur random-matrix benchmark is standard in econophysics and high-dimensional statistics, but its application to test a gravity-model phase transition is novel in this context.
The result contributes an empirical bound: for this universe of 15 large-cap US equities over 2010–2024, the gravity analogy's implicit prediction of low-dimensional (1–2 factor) structure held until approximately 2021, after which a third factor emerged and persisted. Future work could test whether this transition generalizes to other universes (e.g., small-cap, international, or sector-specific portfolios) or whether it is specific to the post-2020 US large-cap regime.
Limitations
Sample size and universe: 15 assets is a small universe. The q-ratio (0.004) is extremely low, giving the Marchenko-Pastur bounds high precision, but the small cross-section limits the generalizability of the phase transition. A larger universe (e.g., 100 or 500 assets) might exhibit a different factor structure or a different transition date.
In-sample only: The per-year factor counts are computed in-sample within each year. We have not tested whether the three-factor structure in 2021–2024 predicts out-of-sample returns, hedging errors, or correlation distances better than a two-factor model. The phase transition is a descriptive fact about the eigenvalue spectrum, not a validated predictive model.
No explicit gravity-model test: The computation tests spectral coherence (factor count over time) but does not directly test the gravity model's predictions. A direct test would require: (a) constructing a predicted correlation matrix from market caps and a gravitational distance function, (b) comparing it to the empirical correlation matrix, and (c) testing whether the prediction error increases after the phase transition. The current result is a necessary condition (breakdown of low-dimensional structure) but not a sufficient test of the gravity analogy.
Factor interpretation: The third factor is inferred from the eigenvalue count but not explicitly characterized. We do not know its loadings or economic interpretation. It could be a sector factor, a volatility factor, a liquidity factor, or a regime-dependent risk dimension. Without loadings, we cannot assess whether it represents a meaningful economic force or a statistical artifact of the post-2020 volatility regime.
Window choice: The 2010–2024 window includes the 2020 COVID-19 shock, which may have induced a structural break unrelated to the gravity model. A robustness check would exclude 2020 or test the transition on a pre-2020 subsample.
Daily frequency: Daily returns may overweight short-term noise relative to the longer-term structural relationships the gravity model aims to capture. Monthly or quarterly returns might yield a different factor structure.
Strengthening the result would require: (a) replication on a larger universe, (b) out-of-sample validation of the three-factor structure's predictive power, (c) explicit construction and testing of the gravity-model predicted correlation matrix, (d) characterization of the third factor's loadings and economic content, and (e) robustness checks on window choice and return frequency. The current result establishes that a phase transition occurred but does not establish its economic cause or its implications for the gravity model's predictive validity.
Research evidence, not investment advice: This is a research finding quantifying the time variation in eigenvalue spectrum structure for a specific universe of large-cap US equities. It is not a trading signal, a recommendation to buy or sell any security, or a forward prediction of returns. The result describes a historical phase transition in spectral coherence; it does not imply that the current three-factor regime will persist or that any hedging strategy based on this structure will be profitable.