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Local constraints play an important role in the effective description of manyquantum systems. Their impact on dynamics and entanglement thermalization arejust beginning to be unravelled. We develop a large $N$ diagrammatic formalismto exactly evaluate the bipartite entanglement of random pure states in largeconstrained Hilbert spaces. The resulting entanglement spectra may beclassified into `phases' depending on their singularities. Our closed solutionfor the spectra in the simplest class of constraints reveals a non-trivialphase diagram with a Marchenko-Pastur (MP) phase which terminates in a criticalpoint with new singularities. The much studied Rydberg-blockaded/Fibonaccichain lies in the MP phase with a modified Page correction to the entanglemententropy, $\Delta S_1 = 0.513595\cdots$. Our results predict the entanglement ofinfinite temperature eigenstates in thermalizing constrained systems andprovide a baseline for numerical studies.

Universal entanglement of typical states in constrained systems