Almost sure uniform convergence of a random Gaussian field conditioned on a large linear form to a non random profile

We investigate the realizations of a random Gaussian field on a finite domain of R^d in the limit where a given linear functional of the field is large. We prove that if its variance is bounded, the field converges uniformly and almost surely to a non random profile depending only on the covariance and the considered linear functional of the field. This is a significant improvement of the weaker L²-convergence in probability previously obtained in the case of conditioning on a large quadratic functional.