Mikhail SODIN

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✬ 2. Spherical ensemble: X = S m (m-dim sphere) Hn,m ⊂ L 2 (S m ) subspace spanned by polynomials in m + 1 variables of degree ≤ n, restricted on S m . Repro-kernel in Hn,m : c(n, m)P P (α,β) n ✫ m m ( 2 , 2 −1) n (x · y), x, y ∈ S m ; Jacobi polynomials of degree n and of index (α, β); i.e., polynomials orthogonal on [−1, 1] with the weight (1 − x) α (1 + x) β . m m m ( Mehler-Heine asymptotics: lim n− 2 2 , 2 P n→∞ −1) z z n cos = n 2 (z) is Bessel’s function, the convergence is locally uniform in C. J m 2 Scaling parameter L = n (the degree) − m 2 J m 2 (z), Limiting spectral measure ρ = Lebesgue measure on the unit ball in R m . 18 ✩ ✪
✬ 3. Kostlan ensemble: Homogeneous polynomials of degree n in m + 1 on X = PR m (m-dim projective space). The scalar product 〈f, g〉 = ✫ f(X) = |J|=n |J|=n fJX J , g(X) = n fJgJ, where J |J|=n J = (j0, j1, j2, ... jm), |J| = j0 + j1 + j2 + ... + jm, n = J gJx J , X J = x j0 0 x j1 1 x j2 2 ... x jm m , n! j0!j1!j2! ... jm! . Complexification: after continuation of the homogeneous polynomials f and g to C m+1 , the scalar product coincides with the one in the Fock-Bargmann space 〈f, g〉 = cm f(Z)g(Z)e −|Z|2 dvol(Z). C m+1 This is the only unitary invariant Gaussian ensemble of homogeneous polynomials. 19 ✩ ✪
Page 1 and 2: ✬ On the number of components of
Page 3 and 4: ✬ ✫ Nodal portrait of a gaussia
Page 5 and 6: ✬ Notation: N(R; F ) the number o
Page 7 and 8: ✬ Part II: Riemannian case X a sm
Page 9 and 10: ✬ Normalization: The functions fL
Page 11 and 12: ✬ Translation-invariant local lim
Page 13 and 14: ✬ Notation: N(fL) the number of c
Page 15 and 16: ✬ Part III. Related works: T.L.Ma
Page 17: ✬ Part IV. Examples 1. Trigonomet
Page 21 and 22: ✬ Part V: Steps in the proof of T
Page 23 and 24: ✬ Step 3: The Kac-Rice bound for

Mikhail SODIN

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