On the relation between the anyon and Calogero models

In order to achieve in configuration space a dimensional reduction from dimension two to one, the lowest Landau level (LLL) projection is not sufficient. One has also, once 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. A thermodynamical argument is proposed which supports this claim. As an illustration, the equivalence of the LLL anyon model in the vanishing magnetic field limit to the Calogero model is established. Some general considerations in favor of an intimate connexion between anyon and Haldane statistics are also given.