Browsing by Subject "harmonic measure"

A convex function, which Hessian determinant equals to one, defined in a convex and bounded polygon is called a surface tension. Moreover at the boundary of the given polygon it is demanded that the function is piece-wise affine. This setting originates from the theory of dimer models but in this thesis the system is studied as such. The boundary condition gives us points of interest i.e. the corners of the polygon and so called quasi-frozen points, where the boundary function is not differentiable. With a suitable homeomorphism one can map the unit disc to the polygon in question. In this setting an explicit formula for the gradient of the surface tension is derived. Furthermore the values of the gradient in corner and quasi-frozen points are derived as limits from, which as a corollary the directed derivatives of the surface tension are studied.