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Fisher zeros and conformality in lattice models

Fisher zeros are the zeros of the partition function in the complex beta=2N_c/g^2 plane. When they pinch the real axis, finite size scaling allows one to distinguish between first and second order transition and to estimate exponents. On the other hand, a gap signals confinement and the method can be used to explore the boundary of the conformal window. We present recent numerical results for 2D O(N) sigma models, 4D U(1) and SU(2) pure gauge and SU(3) gauge theory with N_f=4 and 12 flavors. We discuss attempts to understand some of these results using analytical methods. We discuss the 2-lattice matching and qualitative aspects of the renormalization group (RG) flows in the Migdal-Kadanoff approximation, in particular how RG flows starting at large beta seem to move around regions where bulk transitions occur. We consider the effects of the boundary conditions on the nonperturbative part of the average energy and on the Fisher zeros for the 1D O(2) model.