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We investigate the realizations of a random Gaussian field on a finite domainof ${\mathbb R}^d$ in the limit where a given linear functional of the field islarge. We prove that if its variance is bounded, the field converges uniformlyand almost surely to a non random profile depending only on the covariance andthe considered linear functional of the field. This is a significantimprovement of the weaker $L^2$-convergence in probability previously obtainedin the case of conditioning on a large quadratic functional.

Almost Sure Uniform Convergence of a Random Gaussian Field Conditioned on a Large Linear Form to a Non Random Profile