A team proposes a random sampling method to test the weak Eigenstate Thermalization Hypothesis (wETH), a central conjecture in quantum many-body physics that explains why isolated systems appear to reach thermal equilibrium. The technique is designed specifically for large, sparse Hamiltonians, where exact diagonalization becomes unfeasible.

The Eigenstate Thermalization Hypothesis (ETH) holds that the eigenstates of a chaotic quantum system behave, for local observables, as if they were a thermal ensemble. Its weak version only requires that the fluctuations of expectation values in those eigenstates shrink as the system size grows. Verifying this condition requires exploring Hilbert space dimensions far beyond what exact methods can reach.

To validate the approach, the authors apply it to a well-studied Rydberg atom system that exhibits both two- and three-body dipole-dipole interactions. The random sampling method gives access to system sizes significantly beyond the limit imposed by exact diagonalization, without needing to compute the full spectrum of the Hamiltonian.

The results show that the fluctuations in eigenstate expectation values decay as D⁻¹/², where D is the Hilbert space dimension. This behavior is consistent with the wETH and agrees with previous numerical results, supporting both the validity of the method and the conjecture in this system.