What question did this study set out to answer?

This research aims to determine the minimum number of local observers needed to uniquely identify critical configurations in 2D systems.

March 31, 2026Open Access

Birthday bound for greedy observer identification at 2D critical points: patch collision entropy controls the log L cost

Key Points

This research aims to determine the minimum number of local observers needed to uniquely identify critical configurations in 2D systems.
Establishes a conditional birthday-style upper bound on observer count.
Conjectures lower bounds for specific critical models involving configurations.
Verifies formulas using sampling data across different Potts models and universality classes.
Utilizes Monte Carlo simulations to analyze collision probabilities and critical behavior.
Establishes an upper bound on observer count related to spatial decorrelation and log L cost.
Finds partial activation in ghost sectors verified with high accuracy through repeated Monte Carlo runs.
Confirms the conjectured properties for various Potts models and a tricritical model, demonstrating tightness in certain conditions.

Abstract

**How many independent local observers are required to uniquely identify one of \ (n = Lᵈ \) independent critical configurations? ** We establish a conditional birthday-style upper bound \ (K^* 2d ₂ L / Hₑ₂ () + O (1) \) assuming spatial decorrelation, with a conjectured factor-2 lower bound. For \ (Sq \) -symmetric Potts models at continuous critical points we conjecture greedy tightness \ (\0, 1\ \). Using sampling data for \ (q=2, 3, 4 \) Potts and the BEG tricritical model, we verify the formula across four universality classes. For Ising (\ (q=2 \) ), a Temperley-Lieb algebra decomposition gives an exact orbit count verified to \ (0. 01 \) bits at \ (w=3, 7, 8, 9 \). A quantum extension to measurement-induced phase transitions empirically gives finite \ (K^* \) near the area-law phase. v3 changelog: Corrected w=6 boundary case. Previous versions stated j=3 ghost sector inactive at w=6; 20 independent Monte Carlo runs (L=256+512) show partial activation ω₃ = 0. 415 ± 0. 033 (13. 3σ above zero). The ghost condition 4=0 kills the linear Markov trace but NOT the quadratic collision probability. v4: revised framing (Hᵣ2 non-trivial content = value not boundedness), Theorem→Proposition, birthday probability formula for Δ∈0, 1, DLR corollary for A1

Birthday bound for greedy observer identification at 2D critical points: patch collision entropy controls the log L cost

Key Points

Abstract

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