The Nazarov-Sodin constant and critical points of Gaussian fields - PowerPoint PPT Presentation

The Nazarov-Sodin constant and critical points of Gaussian fields M. McAuley Joint work with Dmitry Beliaev and Stephen Muirhead Mathematical Institute University of Oxford Random Waves in Oxford, June 2018

Preliminaries Let f : R 2 → R be a stationary Gaussian field with zero-mean, unit variance and covariance function κ : R 2 → [ − 1 , 1] and spectral measure ρ , i.e. for x , y ∈ R 2 � R 2 e it · x d ρ ( t ) κ ( x ) = E ( f ( y ) f ( y + x )) = Basic assumptions: 1. κ ∈ C 4+ ( R 2 ) (which implies f ∈ C 2+ ( R 2 ) a.s.) 2. ∇ 2 f (0) is a non-degenerate Gaussian vector We are interested in the geometry of the level sets { f = ℓ } := { x ∈ R 2 | f ( x ) = ℓ } for ℓ ∈ R . RWO2018 NS constant and critical points 2

Previous results For Ω ⊂ R 2 let N LS ( ℓ, Ω) be the number of components of { f = ℓ } in Ω. Theorem (Nazarov-Sodin 2016) If f is ergodic then there exists c NS ( ρ ) ≥ 0 such that N LS (0 , R · Ω) / ( Area (Ω) R 2 ) → c NS ( ρ ) a.s. and in L 1 . Theorem (Kurlberg-Wigman 2018) If ρ has compact support then there exists c NS ( ρ ) ≥ 0 such that E ( N LS (0 , [0 , R ] 2 )) = c NS ( ρ ) R 2 + O ( R ) Moreover c NS ( ρ ) is continuous in ρ (w.r.t. the w ∗ -topology). RWO2018 NS constant and critical points 3

Previous results For Ω ⊂ R 2 let N LS ( ℓ, Ω) be the number of components of { f = ℓ } in Ω. Theorem (Nazarov-Sodin 2016) If f is ergodic then there exists c NS ( ρ, ℓ ) ≥ 0 such that N LS ( ℓ, R · Ω) / ( Area (Ω) R 2 ) → c NS ( ρ, ℓ ) a.s. and in L 1 . Theorem (Kurlberg-Wigman 2018) If ρ has compact support then there exists c NS ( ρ, ℓ ) ≥ 0 such that E ( N LS ( ℓ, [0 , R ] 2 )) = c NS ( ρ, ℓ ) R 2 + O ( R ) Moreover c NS ( ρ, ℓ ) is continuous in ρ (w.r.t. the w ∗ -topology) for each ℓ ∈ R . RWO2018 NS constant and critical points 4

Previous results For Ω ⊂ R 2 let N ES ( ℓ, Ω) be the number of components of { f ≥ ℓ } in Ω. Theorem (Nazarov-Sodin 2016) If f is ergodic then there exists c ES ( ρ, ℓ ) ≥ 0 such that N ES ( ℓ, R · Ω) / ( Area (Ω) R 2 ) → c ES ( ρ, ℓ ) a.s. and in L 1 . Theorem (Kurlberg-Wigman 2018) If ρ has compact support then there exists c ES ( ρ, ℓ ) ≥ 0 such that E ( N ES ( ℓ, [0 , R ] 2 )) = c ES ( ρ, ℓ ) R 2 + O ( R ) Moreover c ES ( ρ, ℓ ) is continuous in ρ (w.r.t. the w ∗ -topology) for each ℓ ∈ R . RWO2018 NS constant and critical points 5

Level sets and excursion sets { f ≥ ℓ } # { Components of { f = ℓ }} ≈ # { Components of { f ≥ ℓ }} + # { Components of { f ≤ ℓ }} Corollary c NS ( ρ, ℓ ) = c ES ( ρ, ℓ ) + c ES ( ρ, − ℓ ) RWO2018 NS constant and critical points 6

Critical points Definition If f is aperiodic we say that a saddle point x is lower connected if it is in the closure of only one component of { f < ℓ } . We say that x is upper connected if it is in the closure of only one component of { f > ℓ } . (When f is periodic, we use a different definition for lower/upper connected saddles.) { f ≥ ℓ 2 } x 2 x 1 { f ≥ ℓ 1 } Figure: x 1 is a lower connected saddle and x 2 is an upper connected saddle. RWO2018 NS constant and critical points 7

Critical points Proposition Let f satisfy the basic assumptions. There exists a function p s − : R → [0 , ∞ ) such that the following holds. Let Ω ⊂ R 2 and let N s − [ ℓ, ∞ ) denote the number of lower connected saddles of f in Ω with level above ℓ . Then � ∞ E [ N s − [ ℓ, ∞ )] = Area(Ω) p s − ( x ) dx . ℓ Analogous statements hold for local maxima, local minima, upper connected saddles and saddles with the densities p m + , p m − , p s + and p s respectively. These functions can be chosen to satisfy p s − + p s + = p s , and such that p m + , p m − and p s are continuous. RWO2018 NS constant and critical points 8

Main results Theorem Let f be a Gaussian field satisfying the basic assumptions, and let p m + , p m − , p s + , p s − denote the critical point densities defined above. Then � ∞ c NS ( ρ, ℓ ) = p m + ( x ) − p s − ( x ) + p s + ( x ) − p m − ( x ) dx (1) ℓ � ∞ p m + ( x ) − p s − ( x ) dx c ES ( ρ, ℓ ) = (2) ℓ and hence c NS and c ES are absolutely continuous in ℓ . In addition c NS and c ES are jointly continuous in ( ρ, ℓ ) provided ρ has a fixed compact support. RWO2018 NS constant and critical points 9

Proof: Intuition Local extrema x 1 x 1 { f ≥ ℓ 1 } { f ≥ ℓ 1 − ǫ } x 2 x 2 { f ≥ ℓ 2 − ǫ } { f ≥ ℓ 2 } { f ≥ ℓ 2 + ǫ } Figure: On raising the level through the local maximum x 1 , the number of level set components decreases by one. On passing through the local minimum x 2 , the number of level set components increases by one. RWO2018 NS constant and critical points 10

Proof: Intuition Lower connected saddle points { f ≥ ℓ 3 } x 3 x 3 { f ≥ ℓ 3 − ǫ } { f ≥ ℓ 3 + ǫ } Figure: On raising the level through the lower connected saddle point x 3 , the number of level set components increases by one. RWO2018 NS constant and critical points 11

Consequences of main results Bounds on c NS and c ES in the isotropic case Proposition (Cheng-Schwartzman 2017) Let f be the random plane wave (RPW) so that κ ( t ) = J 0 ( | t | ) (the 0 -th Bessel function), then � � 1 ( x 2 − 1) e − x 2 2 + e − 3 x 2 p m + ( x ) = p m − ( − x ) = √ 1 x ≥ 0 2 4 2 π 3 / 2 2 π 3 / 2 e − 3 x 2 1 2 . p s ( x ) = √ 4 Substituting these expressions into the main integral equality and considering the number of ‘flip points’ (see Kurlberg-Wigman 2018) shows that Corollary Let f be the RPW and ℓ ≥ 0 , then � √ √ √ 4 π ℓ φ ( ℓ ) ≤ c ES ( ℓ ) ≤ c NS ( ℓ ) ≤ 1 1 � �� 4 π φ ( ℓ ) 2 φ ( 2 ℓ ) + ℓ 2Φ( 2 ℓ ) − 1 RWO2018 NS constant and critical points 12

Consequences of main results Bounds on c NS and c ES in the isotropic case (a) c ES ( ρ, ℓ ) for the RPW (b) c NS ( ρ, ℓ ) for the RPW Figure: Lower bounds (solid) and upper bounds (dashed) for c ES ( ρ, ℓ ) and c NS ( ρ, ℓ ) respectively for the RPW. The bound on c ES ( ρ, ℓ ) for ℓ < 0 is a result of the equality c NS ( ρ, ℓ ) = c ES ( ρ, ℓ ) + c ES ( ρ, − ℓ ) and the fact that c ES ( ρ, ℓ ) is non-decreasing for ℓ < 0 (this part is specific to the RPW). RWO2018 NS constant and critical points 13

Consequences of main results Bounds on c NS and c ES in the isotropic case Similar results hold for all isotropic fields satisfying the basic assumptions. (The general expression for upper and lower bounds becomes more complicated, but depends only on the derivatives of κ at 0.) (a) c ES ( ρ, ℓ ) for the Bargmann-Fock (b) c NS ( ρ, ℓ ) for the Bargmann-Fock field. field. Figure: Lower bounds (solid) and upper bounds (dashed) for c ES ( ρ, ℓ ) and c NS ( ρ, ℓ ) respectively, where ρ is the spectral measure of the Bargmann-Fock field. RWO2018 NS constant and critical points 14

Consequences of main results Derivation of c NS and c ES for 4/5 point spectral measures Proposition Let f be the Gaussian field with spectral measure ρ = αδ 0 + β 2 ( δ K + δ − K ) + γ 2 ( δ L + δ − L ) where β, γ > 0 , α = 1 − β − γ ≥ 0 and K , L ∈ R 2 are linearly independent. Then c NS ( ℓ ) = | K × L | · P ( | Y 1 − Y 2 | ≤ ℓ + X 0 ≤ Y 1 + Y 2 ) , c ES ( ℓ ) = | K × L | · P ( | Y 1 − Y 2 | ≤ | ℓ + X 0 | ≤ Y 1 + Y 2 ) , × denotes the cross product, X 0 ∼ N (0 , α ) , Y 1 ∼ Ray( √ β ) , Y 2 ∼ Ray( √ γ ) and X 0 , Y 1 , Y 2 are independent. If c NS ( ℓ ) � = 0 then N LS , R ( ℓ ) / ( π R 2 ) converges in L 1 to a non-constant random variable and hence does not converge a.s. to a constant, and this statement also holds for c ES and N ES , R ( ℓ ) / ( π R 2 ) . Furthermore p m + ( x ) = p m − ( − x ) = | K × L | · p X 0 + Y 1 + Y 2 ( x ) p s − ( x ) = p s + ( − x ) = | K × L | · p X 0 + | Y 1 − Y 2 | ( x ) where p Z denotes the probability density of a random variable Z. RWO2018 NS constant and critical points 15

Consequences of main results Derivation of c NS and c ES for 4 point spectral measures (a) c ES ( ℓ ) (b) c NS ( ℓ ) Figure: The functions c ES ( ℓ ) (left) and c NS ( ℓ ) (right) with α = 0 for β − γ = 0 (solid), β − γ = 0 . 5 (dashed) and β − γ = 0 . 9 (dotted) respectively. RWO2018 NS constant and critical points 16

Consequences of main results Derivation of c NS and c ES for 5 point spectral measures (a) c ES ( ℓ ) (b) c NS ( ℓ ) Figure: The functions c ES ( ℓ ) (left) and c NS ( ℓ ) (right) with β = γ for α = 0 . 1 (solid), α = 0 . 3 (dashed) and α = 0 . 6 (dotted) respectively. RWO2018 NS constant and critical points 17

Extensions/open questions 1. Characterising p s − (or p s + ) 2. Higher dimensions 3. Continuous differentiability of c NS 4. Bimodality RWO2018 NS constant and critical points 18

References [1] D. Cheng and A. Schwartzman. “Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields”. (2017). [2] P. Kurlberg and I. Wigman. “Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves”. (2018). [3] F. Nazarov and M. Sodin. “Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions”. (2016). RWO2018 NS constant and critical points 19