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We extend the real-space renormalization group (RG) approach to the study ofthe energy level statistics at the integer quantum Hall (QH) transition.Previously it was demonstrated that the RG approach reproduces the criticaldistribution of the {\em power} transmission coefficients, i.e., two-terminalconductances, $P_{\text c}(G)$, with very high accuracy. The RG flow of $P(G)$at energies away from the transition yielded the value of the criticalexponent, $\nu$, that agreed with most accurate large-size lattice simulations.To obtain the information about the level statistics from the RG approach, weanalyze the evolution of the distribution of {\em phases} of the {\emamplitude} transmission coefficient upon a step of the RG transformation. Fromthe fixed point of this transformation we extract the critical level spacingdistribution (LSD). This distribution is close, but distinctively differentfrom the earlier large-scale simulations. We find that away from the transitionthe LSD crosses over towards the Poisson distribution. Studying the change ofthe LSD around the QH transition, we check that it indeed obeys scalingbehavior. This enables us to use the alternative approach to extracting thecritical exponent, based on the LSD, and to find $\nu=2.37\pm0.02$ very closeto the value established in the literature. This provides additional evidencefor the surprising fact that a small RG unit, containing only five nodes,accurately captures most of the correlations responsible for thelocalization-delocalization transition.Comment: 10 pages, 11 figure

Renormalization group approach to energy level statistics at the integer quantum Hall transition