The principal Dirichlet eigenvalue of the sphere minus a high-dimensional orthant decays like a polynomial times 2^{-d}, making the survival exponent of d Brownian particles vanish as d grows.
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On the first hitting time of a high-dimensional orthant
The principal Dirichlet eigenvalue of the sphere minus a high-dimensional orthant decays like a polynomial times 2^{-d}, making the survival exponent of d Brownian particles vanish as d grows.