Graph-theoretic toy models of emergent geometry often rely informally on tensor-network renormalization, entropic gravity, and the ER=EPR correspondence, without being checkable statistical mechanics. 𝐇𝐞𝐫𝐞 𝐰𝐞 𝐫𝐞𝐝𝐮𝐜𝐞 𝐭𝐡𝐞 𝐦𝐨𝐝𝐞𝐥 𝐭𝐨 𝐚 𝐟𝐮𝐥𝐥𝐲 𝐜𝐡𝐞𝐜𝐤𝐚𝐛𝐥𝐞 𝐬𝐭𝐚𝐭𝐢𝐬𝐭𝐢𝐜𝐚𝐥 𝐦𝐞𝐜𝐡𝐚𝐧𝐢𝐜𝐬 𝐬𝐲𝐬𝐭𝐞𝐦. For every edge 𝑒= (𝑢, 𝑣) of a graph we define an 𝐢𝐧𝐭𝐫𝐢𝐧𝐬𝐢𝐜, 𝐛𝐚𝐜𝐤𝐠𝐫𝐨𝐮𝐧𝐝-𝐢𝐧𝐝𝐞𝐩𝐞𝐧𝐝𝐞𝐧𝐭 𝐚𝐜𝐭𝐢𝐨𝐧𝑆 (𝑒) =1+min (𝑑ₐlt (𝑒), Λ), where 𝑑ₐlt (𝑒) is the shortest path between 𝑢 and 𝑣 after deleting 𝑒, and Λ is a fixed cap. We sample the Gibbs measureπ (𝐺) ∝ exp (−𝑆𝐺/ħ) on graphs with fixed (𝑁, 𝐸) via a 𝐝𝐞𝐭𝐚𝐢𝐥𝐞𝐝-𝐛𝐚𝐥𝐚𝐧𝐜𝐞 𝐌𝐞𝐭𝐫𝐨𝐩𝐨𝐥𝐢𝐬 𝐞𝐝𝐠𝐞-𝐫𝐞𝐰𝐢𝐫𝐢𝐧𝐠 𝐜𝐡𝐚𝐢𝐧, with 𝐧𝐨 𝐡𝐚𝐧𝐝-𝐢𝐦𝐩𝐨𝐬𝐞𝐝 𝐬𝐭𝐫𝐮𝐜𝐭𝐮𝐫𝐚𝐥 𝐩𝐫𝐨𝐭𝐞𝐜𝐭𝐢𝐨𝐧𝐬. 𝐊𝐞𝐲 𝐫𝐞𝐬𝐮𝐥𝐭𝐬: • (i) 𝐄𝐱𝐚𝐜𝐭 𝐠𝐫𝐨𝐮𝐧𝐝-𝐬𝐭𝐚𝐭𝐞 𝐟𝐥𝐨𝐨𝐫 𝑆ₘin = 3𝐸 is reproducibly attained• (ii) 𝐇𝐢𝐠𝐡-ħ 𝐥𝐢𝐦𝐢𝐭 → 𝐄𝐫𝐝ős–𝐑é𝐧𝐲𝐢 𝐬𝐭𝐚𝐭𝐢𝐬𝐭𝐢𝐜𝐬• (iii) Finite-size scaling (𝑁=50, 100, 200), unimodal energy histograms, Binder cumulant ≈ 0, and replica edge-overlap ≈ chance 𝐫𝐮𝐥𝐞 𝐨𝐮𝐭 𝐚 𝐜𝐨𝐧𝐭𝐢𝐧𝐮𝐨𝐮𝐬 𝐩𝐡𝐚𝐬𝐞 𝐭𝐫𝐚𝐧𝐬𝐢𝐭𝐢𝐨𝐧• (iv) The system exhibits a 𝐬𝐦𝐨𝐨𝐭𝐡 𝐜𝐫𝐨𝐬𝐬𝐨𝐯𝐞𝐫 at ħ* = 1. 168 ± 0. 031 An adversarial initial condition reveals a 𝐬𝐞𝐜𝐨𝐧𝐝, 𝐞𝐧𝐞𝐫𝐠𝐞𝐭𝐢𝐜𝐚𝐥𝐥𝐲 𝐝𝐞𝐠𝐞𝐧𝐞𝐫𝐚𝐭𝐞 𝐛𝐚𝐬𝐢𝐧: a topologically disconnected dense cluster that is 𝐤𝐢𝐧𝐞𝐭𝐢𝐜𝐚𝐥𝐥𝐲 𝐟𝐫𝐨𝐳𝐞𝐧 𝐚𝐭 𝐥𝐨𝐰 ħ, but becomes accessible above it. A direct control experiment shows: • 𝐋𝐨𝐜𝐚𝐥 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐚𝐥𝐨𝐧𝐞 𝐝𝐨𝐞𝐬 𝐧𝐨𝐭 𝐩𝐫𝐨𝐭𝐞𝐜𝐭 𝐬𝐭𝐫𝐮𝐜𝐭𝐮𝐫𝐞𝐬• Only 𝐭𝐨𝐩𝐨𝐥𝐨𝐠𝐢𝐜𝐚𝐥 𝐝𝐢𝐬𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐢𝐨𝐧 provides stability, and only in the cold regime A two-state mean-field estimate reproduces the 𝐨𝐫𝐝𝐞𝐫 𝐨𝐟 𝐦𝐚𝐠𝐧𝐢𝐭𝐮𝐝𝐞 𝐚𝐧𝐝 𝐬𝐡𝐚𝐩𝐞 of the crossover, but leaves an 𝐞𝐱𝐩𝐥𝐢𝐜𝐢𝐭, 𝐨𝐩𝐞𝐧 𝐭𝐞𝐧𝐬𝐢𝐨𝐧 with the observed behavior. Under the triangle-augmented Forman–Ricci curvature: • The 𝐬𝐩𝐚𝐫𝐬𝐞 𝐛𝐮𝐥𝐤 𝐢𝐬 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞𝐥𝐲 𝐜𝐮𝐫𝐯𝐞𝐝• Dense, kinetically protected regions are 𝐬𝐭𝐫𝐨𝐧𝐠𝐥𝐲 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞𝐥𝐲 𝐜𝐮𝐫𝐯𝐞𝐝 𝐅𝐮𝐥𝐥 𝐜𝐨𝐝𝐞 𝐚𝐧𝐝 𝐫𝐞𝐩𝐫𝐨𝐝𝐮𝐜𝐢𝐛𝐢𝐥𝐢𝐭𝐲 𝐦𝐚𝐭𝐞𝐫𝐢𝐚𝐥𝐬 𝐚𝐫𝐞 𝐢𝐧𝐜𝐥𝐮𝐝𝐞𝐝. Licensed under PolyForm Noncommercial 1. 0. 0, covering all algorithms and implementations across any programming language and hardware platform.
Andres Sebaatian Pirolo (Sun,) studied this question.