Observations reveal that the peaks of the X-ray map and the Sunyaev-Zeldovich (SZ) effect map of some galaxy clusters are offset from each other. In this paper, we perform a set of hydrodynamical simulations of mergers of two galaxy clusters to investigate the spatial offset between the maxima of the X-ray and the SZ surface brightness of the merging clusters. We find that significantly large SZ-X-ray offsets (>100kpc) can be produced during the major mergers of galaxy clusters. The significantly large offsets are mainly caused by a `jump effect occurred between the primary and secondary pericentric passages of the two merging clusters, during which the X-ray peak may jump to the densest gas region located near the center of the small cluster, but the SZ peak remains near the center of the large one. Our simulations show that merging systems with higher masses and larger initial relative velocities may result in larger offset sizes and longer offset time durations; and only nearly head-on mergers are likely to produce significantly large offsets. We further investigate the statistical distribution of the SZ-X-ray offset sizes and find that (1) the number distribution of the offset sizes is bimodal with one peak located at low offsets ~0 and the other at large offsets ~350-450kpc/h, but the objects with intermediate offsets are scarce; and (2) the probabilities of the clusters in the mass range higher than 2x10^{14}Msun/h that have offsets larger than 20, 50, 200, 300, and 500kpc/h are 34.0%, 11.1%, 8.0%, 6.5%, and 2.0% respectively at z=0.7. The probability is sensitive to the underlying pairwise velocity distribution and the merger rate of clusters. Future observations on the offsets for a large number of clusters may put strong constraints on the cosmic velocity fields on the cluster scale and the cluster merger rate. (Abridged)
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