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Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD

We link the spontaneous breakdown of chiral symmetry in Euclidean QCD to the collision of spectral shock waves in the vicinity of zero eigenvalue of Dirac operator. The mechanism, originating from complex Burger's-like equation for viscid, pressureless, one-dimensional flow of eigenvalues, is similar to recently observed weak-strong coupling phase transition in large $N_c$ Yang-Mills theory. The spectral viscosity is proportional to the inverse of the size of the random matrix that replaces the Dirac operator in the universal (ergodic) regime. We obtain the exact scaling function and critical exponents of the chiral phase transition for the averaged characteristic polynomial for $N_c \ge3$ QCD. We reinterpret our results in terms of known properties of chiral random matrix models and lattice data.