Scaling Limits for Mixed Kernels
نویسنده
چکیده
Let μ and ν be measures supported on (−1, 1) with corresponding orthonormal polynomials { pn } and {pn} respectively. Define the mixed kernel K n (x, y) = n−1 ∑ j=0 pμj (x) p ν j (y) . We establish scaling limits such as lim n→∞ π √ 1− ξ √ μ′ (ξ) ν′ (ξ) n K n ( ξ + aπ √ 1− ξ n , ξ + bπ √ 1− ξ n ) = S ( π (a− b) 2 ) cos ( π (a− b) 2 +B (ξ) ) , where S (t) = sin t t is the sinc kernel, and B (ξ) depends on μ, ν and ξ. This reduces to the classical universality limit in the bulk when μ = ν. We deduce applications to the zero distribution of K n , and asymptotics for its derivatives. Orthogonal polynomials, universality limits, scaling limits 42C05
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