We show that there exists a simple generalization of Kazakov's multicritical one-matrix model, which interpolates between the various multicritical points of the model. The associated multicritical potential takes the form of a power series with a heavy tail, leading to a cut of the potential and its derivative at the real axis, and reduces to a polynomial at Kazakov's multicritical points. From the combinatorial point of view the generalized model allows polygons of arbitrary large degrees (or vertices of arbitrary large degree, when considering the dual graphs), and it is the weight assigned to these large order polygons which brings about the interpolation between the multicritical points in the one-matrix model.
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