Random walks, Douglas–Kazakov phase transitions and critical phenomena in topological ensembles

The new critical phenomena in the 2d random walks is found via the relation with the 2d Yang–Mills theory. We also consider the relation between three physical problems: 2D directed lattice random walks in an external magnetic field, ensembles of torus knots and 5d Abelian SUSY gauge theory with massless hypermultiplet in $\Omega$ background. All these systems exhibit the critical behavior typical for the "area $+$ length" statistics of grand ensembles of 2D directed paths. In particular, we have found the new critical behavior in the ensembles of the torus knots and in the instanton ensemble in 5d gauge theory using the combinatorial description. The relation with the integrable model is discussed. Based on joint works with K. Bulycheva and S. Nechaev.