The quasiclassical theory of a nanosize point contacts (PC) between two ferromagnets is developed. The maximum available magnetoresistance values in PC are calculated for ballistic versus diffusive transport through the area of a contact. In the ballistic regime the magnetoresistance in excess of few hundreds percents is obtained for the iron-group ferromagnets. The necessary conditions for realization of so large magnetoresistance in PC, and the experimental results by García et al are discussed.PACS numbers: 74.80. Dm, 74.50.+r, In recent experiments on study of Ni-Ni and Co-Co point contacts (PC), a surprisingly high negative magnetoresistance exceeding 200% has been discovered. 1,2 The set up of the experiment was typical for observation of giant magnetoresistance (GMR), the effect observed earlier in hybrid systems involving ferromagnetic and normal multilayer metals. 3,4 However, for the multilayer structures the typical change of the resistance reached 10% − 30%, which is considerably lower than the corresponding values of Refs. 1,2. So, one can come easily to the conclusion that the main contribution to the MR comes from the region of the PC.A negative magnetoresistance can be due to scattering on domain walls (DW) and this effect has been considered in a number of works 5-8 giving typical values of MR in a range of few percents. Such considerably low values of the MR were obtained assuming that realistic widths of the DW were large, which resulted in a low scattering amplitudes. Considering sharp DW in the ballistic regime one comes to values ∼ 70%. 8 The fact that a sharp DW may give large MR was used in Ref. 2 to explain the anomalously large values of MR in the experiments on the point contacts. 1,2 However, the theory in Ref. 2 is the perturbation theory, it can not be applied to the explanation of 300% effect. The diminishing of the width of DW, when decreasing the size of the constriction was demonstrated by Bruno. 9 The DW width becomes comparable with PC length, and magnetization rotates almost abruptly inside the constriction. This conclusion holds until the diameter of PC is smaller than its actual length. With further increase of constriction size (diameter) the wall will bend outside of PC, and simple energy considerations show that the DW width will be of the order of PC size. 10

We calculate the quantized conductance of nanosize point contacts between two ferromagnets for different mutual orientations of the magnetic moments. It is found that the magnetoresistance MR is a multivalued function of the quantized conductance at the parallel alignment of the magnetizations σ F . This leads us to the conclusion that experimentally observed large fluctuations of MR versus σ F are rather due to the conductance quantization than to measurement errors or a poor reproducibility of the results. Using the results of the calculations we are able to understand experimental data obtained by García et al for MR of the magnetic nanocontacts.