We study the rate of concentration of a Brownian bridge in time one around
the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched
negative sectional curvature, when the distance between the two extremities
tends to infinity. This improves on previous results by A. Eberle, and one of
us. Along the way, we derive a new asymptotic estimate for the logarithmic
derivative of the heat kernel on such manifolds, in bounded time and with one
space parameter tending to infinity, which can be viewed as a counterpart to
Bismut's asymptotic formula in small time
