  The Gaussian entire function is a random entire function, characterised by a
certain invariance with respect to isometries of the plane. We study the
fluctuations of the increment of the argument of the Gaussian entire function
along planar curves. We introduce an inner product on finite formal linear
combinations of curves (with real coefficients), that we call the signed
length, which describes the limiting covariance of the increment. We also
establish asymptotic normality of fluctuations.
