TY - JOUR
T1 - Transportation to random zeroes by the gradient flow
AU - Nazarov, Fedor
AU - Sodin, Mikhail
AU - Volberg, Alexander
N1 - Funding Information:
Keywords and phrases: Zeroes of Gaussian entire functions, transportation, gradient flow, almost independence. AMS Mathematics Subject Classification: 30D99, 60G55 F.N. and A.V. partially supported by the National Science Foundation, DMS grant 0501067. M.S. partially supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grant 357/04.
PY - 2007/9
Y1 - 2007/9
N2 - We consider the zeroes of the random Gaussian entire function f(z)= ∑ σinfk=0ξk zkk! ξ0ξ1,. . . are Gaussian i.i.d. complex random variables) and show that their basins under the gradient flow of the random potential U(z) = log |f(z)| - 1/2|z|2 partition the complex plane into domains of equal area. We find three characteristic exponents 1, 8/5, and 4 of this random partition: the probability that the diameter of a particular basin is greater than R is exponentially small in R; the probability that a given point z lies at a distance larger than R from the zero, it is attracted to decays as e -R8/5 ; and the probability that, after throwing away 1% of the area of the basin, its diameter is still larger than R decays as e -R4. We also introduce a combinatorial procedure that modifies a small portion of each basin in such a way that the probability that the diameter of a particular modified basin is greater than R decays as e -cR4(logR)-3/2.
AB - We consider the zeroes of the random Gaussian entire function f(z)= ∑ σinfk=0ξk zkk! ξ0ξ1,. . . are Gaussian i.i.d. complex random variables) and show that their basins under the gradient flow of the random potential U(z) = log |f(z)| - 1/2|z|2 partition the complex plane into domains of equal area. We find three characteristic exponents 1, 8/5, and 4 of this random partition: the probability that the diameter of a particular basin is greater than R is exponentially small in R; the probability that a given point z lies at a distance larger than R from the zero, it is attracted to decays as e -R8/5 ; and the probability that, after throwing away 1% of the area of the basin, its diameter is still larger than R decays as e -R4. We also introduce a combinatorial procedure that modifies a small portion of each basin in such a way that the probability that the diameter of a particular modified basin is greater than R decays as e -cR4(logR)-3/2.
KW - Almost independence
KW - Gradient flow
KW - Transportation
KW - Zeroes of Gaussian entire functions
UR - http://www.scopus.com/inward/record.url?scp=35148857022&partnerID=8YFLogxK
U2 - 10.1007/s00039-007-0613-z
DO - 10.1007/s00039-007-0613-z
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AN - SCOPUS:35148857022
SN - 1016-443X
VL - 17
SP - 887
EP - 935
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 3
ER -