We study the universal properties of the Lanczos algorithm applied to finite-size many-body quantum systems. Focusing on autocorrelation functions of local operators and on their infinite-time behavior at finite size, we conjecture that in the large-$n$ limit, the ratios between consecutive Lanczos coefficients should have specific scalings with the size of the lattice that we make precise and that depend on the hydrodynamic tail of the autocorrelation function. The scaling associated with strong or approximate zero modes is also discussed. We support our conjecture with a numerical study of different models.

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• Long-range interactions
• Quantum correlations, foundations & formalism
• 1-dimensional systems
• Nonequilibrium lattice models
• Numerical techniques