On the relation between the anyon and Calogero models

In order to achieve a dimensional reduction from dimension two to one not only in phase space but also in configuration space, the lowest Landau level (LLL) projection is not sufficient. One has also, in the LLL, to take the vanishing magnetic field limit, a procedure which can be given a non ambiguous meaning by means of a long distance regulator. As an illustration, the equivalence of the LLL anyon model in the vanishing magnetic field limit to the Calogero model is established. A thermodynamical argument is proposed which supports this claim. Some general considerations in favor of an intimate connexion between anyon and Haldane statistics are also given.