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The statistical correlations of two copies of a d-dimensional elasticmanifold embedded in slightly different frozen disorder are studied using theFunctional Renormalization Group to one-loop accuracy, order O(eps = 4-d).Determining the initial (short scale) growth of mutual correlations, i.e. chaosexponents, requires control of a system of coupled differential (FRG) equations(for the renormalized mutual and self disorder correlators) in a very delicateboundary layer regime. Some progress is achieved at non-zero temperature, wherelinear analysis can be used. A growth exponent a is defined from center of massfluctuations in a quadratic potential. In the case where temperature ismarginal, e.g. a periodic manifold in d=2, we demonstrate analytically andnumerically that a = eps (1/3 - 1/(2 log(1/T)) with interesting and unexpectedlogarithmic corrections at low T. For short range (random bond) disorder ouranalysis indicates that a = 0.083346(6) eps, with large finite sizecorrections.

Chaos in the thermal regime for pinned manifolds via functional RG