SLE and nodal lines. Eugene Bogomolny, Charles Schmit & Rémy Dubertrand. LPTMS, Orsay, France. SLE and nodal lines p.

Transcription

1 SLE and nodal lines p. SLE and nodal lines Eugene Bogomolny, Charles Schmit & Rémy Dubertrand LPTMS, Orsay, France

2 SLE and nodal lines p. Outline Motivations Loewner Equation Stochastic Loewner Equation (SLE) Our work and our preliminar results Conclusion

3 SLE and nodal lines p. Motivations For every classically chaotic system, we want to know properties of eigenvalues and eigenfunctions solutions of Schrodinger s equation : (H E)Ψ = (1) For chaotic two dimensionnal billiards, eigenfunctions have the same properties as chaotic wave function (Berry, J. Phys. A, 12, 283, 1977): Ψ( x) = m ZC m φ m ( x) (2) C m = C m : Gaussian random variables (φ m ) : an orthogonal complete (real) basis of the Hilbert space. Blum, Gnutzmann and Smilansky (Phys. Rev. Lett. 88, 11411, 22) initiated the investigation of nodal domains of wave functions for different systems.

4 SLE and nodal lines p. Chaotic billard and random wave function Nodal domains of stadium with E = Gaussian random wave function with k = 1 According to Bogomolny & Schmit (Phys. Rev. Lett. 88, 11412, 22), nodal domains of chaotic wave functions are well described by a percolation-like model.

5 SLE and nodal lines p. From percolation to conformal maps Consider the site percolation on triangular lattice. We can see it with a hexagonal lattice : each hexagon is coloured in black with probability p or in white in probability 1 p. One can follow the frontier between black and white clusters. What is the scaling limit of such a line at critical threshold p = p c? It is commonly assumed that this scaling limit is in two dimensions conformally invariant. We are now going to build such lines using conformal maps.

6 SLE and nodal lines p. Simple curve and conformal maps Let C be a simple curve in H, parametrized by γ : R + C By Riemann s theorem there exists a conformal map g t : H\γ([;t]) H. g t is not unique, can be composed with a conformal self-map of H (3 free real parameters). We choose normalization such that g t is unique : g t (z) = z + 2t z +... when z in a convenient parametrization, one can call t conformal time such that we have some additivity : g t g s = g t+s

7 SLE and nodal lines p. The Loewner equation As an example, let τ be a small positive real. For z >> γ(τ), γ([;τ]) looks like a little vertical line from to 2i τ. In this case, we shall write the conformal map explicitly : g τ (z) = z 2 + 4τ z + 2τ z +... when z >> τ For fixed t >, let define U(t) = g t (γ(t)) the image of the tip. g t+τ (z) = g t (z) + dg t dt (z) = 2 g t (z) U(t) 2τ g t (z) U(t) +... when z >> γ(τ) with g (z) = z (3) This is the Loewner equation. U(t) is called forcing function. C is called the trace of U(t). We can extend the result when C is not a simple curve.

8 SLE and nodal lines p. Examples : what shape for C when U(t) simple? 2i t z²+4t Linear forcing U(t) = t Trace for linear forcing

9 SLE and nodal lines p. Other simple forcing Square root forcing U(t) = 2 k(1 t) 2 k, k = 1, 2, 2.5, 3, 3.5, 4, 5, Trace for square forcing

10 SLE and nodal lines p. 1 Reverse way : What forcing for simple curve? very few exact solution for fixed trace : a slit with an angle α : 1 2α U(t) = 2 t α(1 α) g t (z) = ( αt ) 1 α ( (1 α)t z + 2 z 2 1 α α ) α gt α

11 SLE and nodal lines p. 1 Another exact case an arc of circle from to a(t) : a(t) gt g a(t) (z) = b 3 b 2 + c 2 F a(t) (z) (b 2 + c 2 ) + 2b3 + 3bc 2 2(b 2 + c 2 ) ( z ) 2 F a(t) (z) = + c 2 1 z/b where b = a 2 Re a and c = a 2 Im a a and t linked by t = (Re a) 2 /8 + (Im a) 2 /4.

12 SLE and nodal lines p. 1 Example of (non solvable) forcing for a simple trace Sinusoidal trace x = 1 cosy Forcing function for sinusoidal trace

13 SLE and nodal lines p. 1 From Loewner equation to SLE Now let take a brownian motion for the forcing function : < U(t)U(s) >= κ t s Ensemble of the generated traces : SLE κ Main properties : κ < 4 : the trace is a simple curve κ 4 : self-intersections can occur κ 8 : the trace fills a region of H

14 SLE and nodal lines p. 1 Examples of SLE κ κ = κ = 6 Simulations done with fast sle (T. Kennedy, 25) κ = 25

15 SLE and nodal lines p. 1 SLE and discrete models the Self Avoiding Walk ensemble converges at the scaling limit to SLE 8/3. the Loop Erased Random Walk is obtained with a random walk where one erase its loop in chronogical order. Schramm proved that, assuming that the scaling limit exists and is conformally invariant, this limit is the SLE 2 ensemble. in the site percolation on a triangular lattice, one can consider lines between cluster; they can be viewed as lines on a hexagonal lattice. Smirnov proved that the scaling limit of these lines at the critical threshold is the SLE 6 ensemble.

16 SLE and nodal lines p. 1 Our work We consider nodal lines of random waves functions. Assuming the percolation-like model, we want to know whether it looks like traces from SLE Nodal line of random wave function Forcing function

17 SLE and nodal lines p. 1 Details of precedent curves Nodal line of random wave function Forcing function

18 SLE and nodal lines p. 1 Our preliminar results We computed variance for 1 lines computed with E = k 2 = 5 2 (fit in red is y = 6x ) : The saturation may be due to a finite size effect.

19 SLE and nodal lines p. 1 More detailed curves,2-1,15-2,1-3 -4,5-5 -6,5,1,15,2,25,3 Variance as a function of t, the fit is y = 4.3x Points inside a bin, the fit gives σ 2 = 7.4

20 SLE and nodal lines p. 2 Conclusions preliminar results show no contradiction between random wave model and SLE 6, this encourages the description of random wave function with a percolation-like model. we can use this method to another ensemble of functions with different correlation function.