Hexagon remainder function in the limit of self-crossing up to three loops

We consider Wilson loops in planar N=4 SYM for null polygons in the limit of two crossing edges. The analysis is based on a renormalisation group technique. We show that the previously obtained result for the leading and next-leading divergent term of the two loop hexagon remainder is in full agreement with the appropriate continuation of the exact analytic formula for this quantity. Furthermore, we determine the coefficients of the leading and next-leading singularity for the three loop remainder function for null n-gons with n >= 6.