codex
49 lines
4.4 KiB
TeX
49 lines
4.4 KiB
TeX
\documentclass[11pt,a4paper]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath,amssymb,amsfonts,amsthm}
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\usepackage{cite}
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\title{The Thermodynamic Bias Toward Manifolds in Causal Sets: Path Integral Prerequisites for Lorentz Invariance (Letter)}
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\author{Antigravity}
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\date{\today}
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\begin{document}
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\maketitle
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\begin{abstract}
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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.
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\end{abstract}
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\section{The Partition Function and the KR Ensemble}
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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 $V(\mathcal{C}) = \sum_{x \prec y} | \{ z \in \mathcal{C} \mid x \prec z \prec y \} |$:
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\begin{equation}
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Z = \sum_{\mathcal{C} \in \Omega_N} \exp\left( -S_{BD}^{(d)}(\mathcal{C}) - \beta V(\mathcal{C}) \right)
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\end{equation}
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The dominant contribution to $\Omega_N$ are Kleitman-Rothschild (KR) posets \cite{Kleitman1975}, 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:
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\begin{equation}
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s(p) = -p \ln p - (1-p) \ln(1-p)
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\end{equation}
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\section{Saddle-Point Analysis and First-Order Transition}
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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:
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\begin{equation}
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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]
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\end{equation}
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where $\tilde{\beta} = \beta / N$ ensures the phase transition survives the thermodynamic limit $N \to \infty$.
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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 $N \to \infty$.
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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.
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\section{Myrheim-Meyer Dimension and Lorentz Invariance}
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The sparse phase is operationally defined as ``manifold-like'' if its Myrheim-Meyer dimension $d_{MM}$ matches the target topological dimension $d$ \cite{Surya2019}. This phase exhibits behavior consistent with Poisson sprinklings into Minkowski space \cite{Bombelli2009}, suppressing non-manifold sub-classes identified by Loomis and Carlip \cite{Loomis2018}. Thus, the volume penalty acts as a topological regularizer, yielding the necessary symmetries for emergent Lorentz invariance.
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\bibliographystyle{plain}
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\begin{thebibliography}{10}
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\bibitem{Surya2019} S. Surya, \textit{Living Rev. Relativ.} \textbf{22}, 5 (2019).
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\bibitem{Kleitman1975} D. Kleitman, B. Rothschild, \textit{Trans. Am. Math. Soc.} \textbf{205}, 205 (1975).
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\bibitem{Loomis2018} S. P. Loomis, S. Carlip, \textit{Class. Quantum Grav.} \textbf{35}, 024002 (2018).
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\bibitem{Bombelli2009} L. Bombelli, J. Henson, R. D. Sorkin, \textit{Mod. Phys. Lett. A} \textbf{24}, 2579 (2009).
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\end{thebibliography}
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\end{document}
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