Anisotropic exceptional points of arbitrary order

A pair of anisotropic exceptional points (EPs) of arbitrary order is found in a class of non-Hermitian random systems with asymmetric hoppings. Both eigenvalues and eigenvectors exhibit distinct behaviors when these anisotropic EPs are approached from two orthogonal directions in the parameter space. For an order-N anisotropic EP, the critical exponents ν of phase rigidity are (N-1)/2 and N-1, respectively. These exponents are universal within the class. The order-N anisotropic EPs split and trace out multiple ellipses of EPs of order 2 in the parameter space. For some particular configurations, all the EP ellipses coalesce and form a ring of EPs of order N. Crossover to the conventional order-N EPs with ν=(N-1)/N is discussed.