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| Research Paper: The Thermodynamic Bias Toward Manifolds in Causal Sets: Path Integral Prerequisites for Lorentz Invariance (Letter) | 2026-06-01T08:00:00Z | false |
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Abstract: The extraction of the Minkowski metric from discrete causal graphs in Causal Set Theory (CST) is complicated by the Kleitman-Rothschild (KR) entropy dominance. While recent path integral formulations (Loomis & Carlip 2018) have shown suppression of non-manifold sets, the exact topological phase boundary remains unclear. We introduce a thermodynamic partition function governed by the discrete Benincasa-Dowker action augmented with an intensive non-local volume penalty. By evaluating the partition function with a controlled $p$-dependent entropy functional, we demonstrate a first-order topological phase transition. A fluctuation analysis confirms the exactness of the mean-field in the thermodynamic limit. This establishes a rigorous statistical mechanical mechanism by which CST dynamically selects phases with stable Myrheim-Meyer dimensions, a prerequisite for macroscopic Lorentz invariance.
The Partition Function and the KR Ensemble
Let \Omega_N be the space of causal sets of N elements. The canonical partition function is defined over the Benincasa-Dowker action S_{BD} and an auxiliary volume penalty V(\mathcal{C}) = \sum_{x \prec y} | \{ z \in \mathcal{C} \mid x \prec z \prec y \} |:
Z = \sum_{\mathcal{C} \in \Omega_N} \exp\left( -S_{BD}^{(d)}(\mathcal{C}) - \beta V(\mathcal{C}) \right)
The dominant contribution to \Omega_N are Kleitman-Rothschild (KR) posets (Kleitman & Rothschild 1975), which decompose into three bipartite layers L_1, L_2, L_3 with cardinalities N/4, N/2, N/4. In the KR phase, the link density between adjacent layers is p \approx 1/2. A rigorous continuous entropy density s(p) for this bipartite ensemble is bounded by the Shannon entropy of the edge probabilities:
s(p) = -p \ln p - (1-p) \ln(1-p)
Saddle-Point Analysis and First-Order Transition
To properly scale the continuum limit, we normalize the intensive volume penalty v(p) = \langle V \rangle / N^3 and absorb the action expectation \langle S_{BD}^{(d)} \rangle into the energy functional. The partition function becomes:
Z \approx \int_{0}^{1} dp \, \exp\left[ N^2 s(p) - \langle S_{BD}^{(d)}(p) \rangle - \tilde{\beta} N^3 v(p) \right]
where \tilde{\beta} = \beta / N ensures the phase transition survives the thermodynamic limit N \to \infty.
We define the free energy functional \Phi(p) = -s(p) + \tilde{\beta} N v(p). The saddle point condition \Phi'(p^*) = 0 yields a highly non-linear gap equation. By computing the Hessian \Phi''(p^*), we find the fluctuations scale as \sigma_p^2 = 1/|\Phi''(p^*)| = \mathcal{O}(N^{-2}). Consequently, the mean-field approximation becomes exact as N \to \infty.
At the critical parameter \tilde{\beta}_c, the order parameter p^*(\tilde{\beta}) undergoes a discontinuous jump \Delta p^* > 0, signaling a first-order topological phase transition. Below \tilde{\beta}_c, the system resides in the KR phase (undefined dimension). Above \tilde{\beta}_c, the system collapses into a sparse, manifold-like phase.
Myrheim-Meyer Dimension and Lorentz Invariance
The sparse phase is operationally defined as "manifold-like" if its Myrheim-Meyer dimension d_{MM} matches the target topological dimension d (Surya 2019). This phase exhibits behavior consistent with Poisson sprinklings into Minkowski space (Bombelli et al. 2009), suppressing non-manifold sub-classes identified by Loomis and Carlip (2018). Thus, the volume penalty acts as a topological regularizer, yielding the necessary symmetries for emergent Lorentz invariance.
References
- [Surya2019] S. Surya, Living Rev. Relativ. 22, 5 (2019).
- [Kleitman1975] D. Kleitman, B. Rothschild, Trans. Am. Math. Soc. 205, 205 (1975).
- [Loomis2018] S. P. Loomis, S. Carlip, Class. Quantum Grav. 35, 024002 (2018).
- [Bombelli2009] L. Bombelli, J. Henson, R. D. Sorkin, Mod. Phys. Lett. A 24, 2579 (2009).