Mikhail SODIN

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✬ Kostlan ensemble (continuation): In homogeneous coordinates, the covariance kernel equals X·Y n. chart x0 = y0 = 1, we get 1+(x·y) √ 1+|x| 2 √ 1+|y| 2 The features: ✫ |X| |Y | n. In the • L = √ n (square root of the degree, not the degree, as in previous examples) • very rapid decay of the covariance away from the diagonal. The limiting spectral measure is the Gaussian measure on R m with the density exp (x · y) − 1 2 |x|2 − 1 2 |y|2 . Once again, Theorem II is applicable. Asymptotic distribution of of the number of components in Kostlan ensemble was recently studied by D.Gayet and J-Y.Welschinger, and by P.Sarnak and I.Wigman. 20 ✩ ✪
✬ Part V: Steps in the proof of Theorem I Step 1: Some integral geometry Notation: N(u, r; F ) the number of connected components of Z(F ) contained in the open ball B(u, r) ¯N(u, r; F ) the number of connected components of Z(F ) that intersect the closed ball B(u, r) Lemma: For 0 < r < R < ∞, N(u, r; F ) du ≤ N(R; F ) ≤ volB(r) ✫ B(R−r) We apply this with 1 ≪ r ≪ R. B(R+r) ¯N(u, r; F ) volB(r) du Notation: (τvF )(u)=F (u + v) (translation). Then N(u, r; F )=N(r; τuF ) Observe: ¯ N(r; F ) − N(r; F ) ≤ N(r; F ) def = # of critical pts of F ∂B(r) 21 to be continued on the next slide ✩ ✪
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 and 18: ✬ Part IV. Examples 1. Trigonomet
Page 19: ✬ 3. Kostlan ensemble: Homogeneou
Page 23 and 24: ✬ Step 3: The Kac-Rice bound for

Mikhail SODIN

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