Real roots of random polynomials and zero crossing properties of diffusion equation

Transcription

1 Real roots of random polynomials and zero crossing properties of diffusion equation Grégory Schehr Laboratoire de Physique Théorique et Modèles Statistiques Orsay, Université Paris XI G. S., S. N. Majumdar, Phys. Rev. Lett. 99, (2007), arxiv: , J. Stat. Phys. 132, (2008), arxiv: Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 1 / 39

2 Introduction Persistence probability p 0 (t) X(t) stochastic random variable evolving in time t, X(t) = 0 Persistence probability p 0 (t) Proba. that X has not changed sign up to time t X 0 t time Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 2 / 39

3 Introduction Persistence probability p 0 (t) X(t) stochastic random variable evolving in time t, X(t) = 0 Persistence probability p 0 (t) Proba. that X has not changed sign up to time t Persistence in spatially extended systems phase ordering kinetics Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 2 / 39

4 Introduction: Phase ordering kinetics Glauber dynamics of 2d Ising model at T = 0, H Ising = J i,j σ iσ j σ i = ±1 t 1 = 0 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 3 / 39

5 Introduction: Phase ordering kinetics Glauber dynamics of 2d Ising model at T = 0, H Ising = J i,j σ iσ j σ i = ±1 t 1 = 0 t 2 = 10 2 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 3 / 39

6 Introduction: Phase ordering kinetics Glauber dynamics of 2d Ising model at T = 0, H Ising = J i,j σ iσ j σ i = ±1 t 1 = 0 t 2 = 10 2 t 3 = 10 4 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 3 / 39

7 Introduction: Phase ordering kinetics Glauber dynamics of 2d Ising model at T = 0, H Ising = J i,j σ iσ j σ i = ±1 t 1 = 0 t 2 = 10 2 t 3 = 10 4 t 4 = 10 6 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 3 / 39

8 Introduction Persistence probability p 0 (t) X(t) stochastic random variable evolving in time t, X(t) = 0 Persistence probability p 0 (t) Proba. that X has not changed sign up to time t Persistence in spatially extended systems phase ordering kinetics ( 94-) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 4 / 39

9 Introduction Persistence probability p 0 (t) X(t) stochastic random variable evolving in time t, X(t) = 0 Persistence probability p 0 (t) Proba. that X has not changed sign up to time t Persistence in spatially extended systems phase ordering kinetics ( 94-) diffusion field ( 96-) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 4 / 39

10 Introduction Persistence probability p 0 (t) X(t) stochastic random variable evolving in time t, X(t) = 0 Persistence probability p 0 (t) Proba. that X has not changed sign up to time t Persistence in spatially extended systems phase ordering kinetics ( 94-) diffusion field ( 96-) height of a fluctuating interface ( 97-)... Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 4 / 39

11 Introduction Persistence probability p 0 (t) X(t) stochastic random variable evolving in time t, X(t) = 0 Persistence probability p 0 (t) Proba. that X has not changed sign up to time t Persistence in spatially extended systems phase ordering kinetics ( 94-) diffusion field ( 96-) height of a fluctuating interface ( 97-)... p 0 (t) t θ p A. J. Bray, S. N. Majumdar, G. S., Adv. Phys. 62, pp (2013), arxiv: Persistence and First-Passage Properties in Non-equilibrium Systems Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 4 / 39

12 Motivations : persistence for the diffusion equation Diffusion equation with random initial conditions t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 5 / 39

13 Motivations : persistence for the diffusion equation Diffusion equation with random initial conditions t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) Diffusion equation (or heat equation) is universal and ubiquitous in nature Ordering dynamics for O(N)-symmetric spin models in the limit N see A. Dembo, S. Mukherjee 12 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 5 / 39

14 Motivations : persistence for the diffusion equation Diffusion equation with random initial conditions t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) Single length scale l(t) t 1/2 Diffusion equation (or heat equation) is universal and ubiquitous in nature Ordering dynamics for O(N)-symmetric spin models in the limit N see A. Dembo, S. Mukherjee 12 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 5 / 39

15 Motivations : persistence for the diffusion equation Diffusion equation with random initial conditions t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) Single length scale l(t) t 1/2 Persistence p 0 (t, L) for a d-dim. system of linear size L p 0 (t, L) Proba. that φ(x, t) has not changed sign up to t S. N. Majumdar, C. Sire, A. J. Bray and S. J. Cornell, PRL 96 B. Derrida, V. Hakim and R. Zeitak, PRL 96 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 5 / 39

16 Motivations : persistence for the diffusion equation Diffusion equation with random initial conditions t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) Single length scale l(t) t 1/2 Persistence p 0 (t, L) for a d-dim. system of linear size L p 0 (t, L) Proba. that φ(x, t) has not changed sign up to t t θ(d) L = 32 L = 64 L = 128 L = 256 p 0 (t,l) L 2θ(d) t Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 5 / 39

17 Motivations : persistence for the diffusion equation Diffusion equation with random initial conditions t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) Single length scale l(t) t 1/2 Persistence p 0 (t, L) for a d-dim. system of linear size L p 0 (t, L) Proba. that φ(x, t) has not changed sign up to t p 0 (t, L) L 2θ(d) h(t/l 2 ) θ(1) = θ(2) = , Numerics Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 5 / 39

18 Motivations : persistence for the diffusion equation Diffusion equation with random initial conditions t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) A. Watson, Science 96 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 5 / 39

19 Persistence in 1d diffusion : NMR experiments on Xe p 0 (t, L) t θexp(1) θ exp (1) 0.12 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 6 / 39

20 Motivations : Real random polynomials n 1 K n (x) = a i x i i=0 Real Kac s polynomials a i Gaussian random variables, a i = 0, a i a j = δ ij Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 7 / 39

21 Motivations : Real random polynomials n 1 K n (x) = a i x i i=0 Real Kac s polynomials a i Gaussian random variables, a i = 0, a i a j = δ ij Complex roots Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 7 / 39

22 Motivations : Real random polynomials n 1 K n (x) = a i x i i=0 Real Kac s polynomials a i Gaussian random variables, a i = 0, a i a j = δ ij Real roots N n mean number of roots on the real axis M.Kac 43 N n 2 π log n Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 7 / 39

23 Motivations : Real roots of Kac s polynomials q 0 (n) Probability that K n (x) has no real root in [0, 1] q 0 (n) n γ with γ = 0.19(1) (Numerics) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 8 / 39

24 Purpose : a link between random polynomials & diffusion equation n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 Generalized Kac s polynomials a i Gaussian random variables, a i = 0, a i a j = δ ij Proba. of no real root q 0 (n) n b(d) Persistence of diffusion p 0 (t, L) L 2θ(d) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 9 / 39

25 Purpose : a link between random polynomials & diffusion equation n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 Generalized Kac s polynomials a i Gaussian random variables, a i = 0, a i a j = δ ij Proba. of no real root q 0 (n) n b(d) Persistence of diffusion p 0 (t, L) L 2θ(d) b(d) = θ(d) G. S., S. N. Majumdar 07 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 9 / 39

26 Purpose : a link between random polynomials & diffusion equation n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 Generalized Kac s polynomials a i Gaussian random variables, a i = 0, a i a j = δ ij Proba. of no real root q 0 (n) n b(d) Persistence of diffusion p 0 (t, L) L 2θ(d) b(d) = θ(d) G. S., S. N. Majumdar 07 A. Dembo, S. Mukherjee 12 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July 14 9 / 39

27 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

28 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

29 Persistence of diffusion equation t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) φ(x, t) = y <L d d yg(x y, t)φ(y, 0) G(x, t) = (4πt) d 2 exp ( x 2 /4t) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

30 Persistence of diffusion equation t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) φ(x, t) = y <L d d yg(x y, t)φ(y, 0) G(x, t) = (4πt) d 2 exp ( x 2 /4t) Mapping of φ(x, t) to a Gaussian stationary process 1 Normalized process X(t) = φ(x,t) φ(x,t) 2 1/2 ( ) d X(t)X(t 4 ) tt 4, t, t L 2 (t+t ) 2 1, t, t L 2 2 New time variable T = log t, for t L 2 X(T )X(T ) = [ cosh((t T )/2) ] d/2 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

31 Persistence for a Gaussian stationary process (GSP) X(T ) is a GSP with correlations X(T )X(T ) = a(t T ) a(t ) = (cosh(t /2)) d/2 Persistence probability P 0 (T ) (by Slepian s lemma) For T 1 a(t ) exp ( d 2 T ) P 0 (T ) exp ( θ(d)t ) Reverting back to t = exp (T ) p 0 (t, L) t θ(d) 1 t L 2 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

32 Persistence of diffusion equation Normalized process X(t) = φ(x,t) φ(x,t) 2 1/2 ( ) d X(t)X(t 4 ) tt 4, t, t L 2 (t+t ) 2 1, t, t L 2 t θ(d) L = 32 L = 64 L = 128 L = 256 p 0 (t,l) L 2θ(d) t Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

33 Persistence of diffusion equation Normalized process X(t) = φ(x,t) φ(x,t) 2 1/2 ( ) d X(t)X(t 4 ) tt 4, t, t L 2 (t+t ) 2 1, t, t L 2 p 0 (t, L) L 2θ(d) h(t/l 2 ) h(u) { u u θ(d), u 1 u c st, u 1 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

34 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

35 Real roots of generalized Kac s polynomials n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 Averaged density of real roots for n ρ (x) = (Li 1 d/2(x 2 )(1 + Li 1 d/2 (x 2 )) Li 2 d/2 (x 2 )) 1 2 π x (1 + Li 1 d/2 (x 2 )) ρ(x) x Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

36 Real roots of generalized Kac s polynomials n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

37 Real roots of generalized Kac s polynomials n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 Mean number of real roots in [0, 1] : Kac-Rice formula N n [0, 1] = 1 0 ρ n (x) dx 1 d 2π 2 log n Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

38 Probability of no real root for K n (x) 3 ρ x Dembo et al. 02 Statistical independence of K n (x) in the 4 sub-intervals = Focus on the interval [0, 1] P 0 (x, n) Proba. that K n (x) has no real root in [0, x] Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

39 Probability of no real root for K n (x) Two-point correlator n 1 C n (x, y) = K n (x)k n (y) = i (d 2)/2 (xy) i i=0 Normalization C n (x, y) C n (x, y) = (C n (x, x)) 1 2 (C n (y, y)) 1 2 Change of variable ρ x = 1 1 t, t x Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

40 Probability of no real root for K n (x) Scaling limit Normalized correlator in the scaling limit t 1, n 1 keeping t = t n fixed C n (t, t ) C( t, t ) with the asymptotic behaviors ( ) t t d 4 C( t, t 4, t, t ) ( t + t 1 ) 2 1, t, t 1 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

41 Persistence of diffusion equation (reminder) t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) φ(x, t) = d d yg(x y)φ(y, 0) G(x, t) = (4πt) d 2 exp ( x 2 /4t) Mapping of φ(x, t) to a Gaussian stationary process 1 Normalized process X(t) = φ(x,t) φ(x,t) 2 1/2 ( tt ) d 4 X(t)X(t 4, t, t ) (t + t ) 2 L 2 1, t, t L 2 2 Persistence probability p 0 (t, L) p 0 (t, L) L 2θ(d) h(t/l 2 ) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

42 Probability of no real root for K n (x) ( ) C( t, t ) 4 t t d 4, t, t 1 ( t+ t ) 2 1, t, t 1 P 0 (x, n) Proba. that K n (x) has no real root in [0, x] Scaling form for P 0 (x, n) P 0 (x, n) n θ(d) h(n(1 x)) h(u) { c st, u 1 u θ(d), u 1 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

43 Probability of no real root for K n (x) Scaling form for P 0 (x, n) P 0 (x, n) n θ(d) h(n(1 x)) h(u) { c st, u 1 u θ(d), u 1 q 0 (n) Probability that K n (x) has no real root in [0, 1] q 0 (n) = P(1, n) n θ(d) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

44 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

45 Numerical check of the scaling form Numerical computation of P 0 (x, n) for d = 2 n = 128 n = 256 n = 512 n = 1024 P 0 (x,n) 1e x Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

46 Numerical check of the scaling form Numerical computation of P 0 (x, n) for d = 2 P 0 (x,n) n θ(2) n(1-x) n = 128 n = 256 n = 512 n = 1024 P 0 (x, n) n θ(d) h(n(1 x)) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

47 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

48 Conclusion A link between diffusion equation and random polynomials Proba. of no real root q 0 (n) n b(d) Persistence of diffusion p 0 (t, L) L 2θ(d) b(d) = θ(d) 1 Universality see A. Dembo, S. Mukherjee 12 2 Towards exact results for θ(d), 1/(4 3) θ(2) 1/4 G. Molchan 12 W. Li, Q. M. Shao 02 see also D. Zaporozhets 06 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

49 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

50 Generalization to k zero crossings Diffusion equation t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) p k (t, L) Proba. that φ(x, t) crosses zero k times up to t Real polynomials n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 q k (n) Proba. that K n (x) has exactly k real roots Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

51 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

52 Mean field approximation for diffusion equation For t L 2, p k (t, L) is given by P k (T ), T = log t P k (T ) Proba. that X(T ) crosses zero k times up to T X(T )X(T ) = (cosh (T T )) d/2 = 1 d 16 T 2 + o(t 2 ) N (T ) Mean number of zero crossings in [0, T ] = ρt ρ = 1 d 2π 2 Assuming that the zeros of X(T ) are independent P k (T ) = (ρt )k e ρt k! Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

53 Mean field approximation for diffusion equation For t L 2, p k (t, L) is given by P k (T ), T = log t P k (T ) Proba. that X(T ) crosses zero k times up to T Assuming that the zeros of X(T ) are independent P k (T ) = (ρt )k e ρt k! For k 1, T 1, k/ρt fixed ( ) k log P k (T ) T ϕ ρt, ϕ(x) = ρ(x log x x + 1) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

54 Scaling form for a smooth GSP X(T ) is a GSP with X(T )X(T ) = [sech((t T )/2)] d/2 P k (T ) Proba. that X(T ) crosses 0 exactly k times up to T ( ) k log P k (T ) T ϕ ρt ϕ(x) is a large deviation function Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

55 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

56 A more refined analysis Generating function S.N. Majumdar, A.J. Bray, PRL 98 ˆP(z, T ) = z k P(k, T ) exp( ˆθ(z)T ) k=0 where ˆθ(z) depends continuously on z Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

57 Application to the integrated Brownian motion Integrated Brownian motion (Random acceleration process) d 2 x(t) dt 2 = ζ(t), ζ(t)ζ(t ) = δ(t t ) Generating function T.W. Burkhardt 01 ˆP(z, T ) = ˆθ(z) = 1 4 z k P(k, T ) exp( ˆθ(z)T ) k=0 ( 1 6 π ArcSin ( z 2) ), 0 z 2 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

58 Application to the integrated Brownian motion Integrated Brownian motion (Random acceleration process) d 2 x(t) dt 2 = ζ(t), ζ(t)ζ(t ) = δ(t t ) Exact result G. S., S. N. Majumdar 08 ( ) p k (t) t ϕ k ρ log t 3, ρ = 2π ( ) 3 2x ϕ(x) = 2π x log + 1 (1 6π ( )) x ArcSin x x Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

59 Probability of k-zero crossings for diffusion equation t φ(x, t) = 2 φ(x, t) φ(x, 0)φ(x, 0) = δ d (x x ) p k (t, L) Proba. that φ(x, t) crosses zero k times up to t ( ) k p k (t, L) t ϕ log t, t L 2 Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

60 Numerical check log p k(t, L) log t as a function of k -log(p k (t,l))/log(t) t=256 t=512 t=1024 t= k Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

61 Numerical check log p k(t, L) log t as a function of k log t log(p k (t,l))/log(t) t= t=512 t=1024 t= πk/log(t) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

62 Outline 1 Mapping to a Gaussian Stationary Process (GSP) The case of diffusion equation The case of random polynomials Numerical check Conclusion 2 Probability of k-zero crossings Generalization to k zero crossings for diffusion and polynomials Mean field approximation and large deviation function A more refined analysis Conclusion Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

63 Conclusion Large deviation function for random polynomials n 1 K n (x) = a 0 + a i i (d 2)/4 x i i=1 q k (n) Proba. that K n (x) has exactly k real roots in [0, 1] ( ) q k (n) n ϕ k log n Extension to other class of polynomials Extension to real eigenvalues of real random matrices (Ginibre s matrices) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

64 A heuristic argument Diffusion equation φ(x = 0, t) = (4πt) d/2 = S 1/2 d (4πt) d/2 0< x <L L Ψ(r) = S 1/2 d r 1 2 (d 1) lim r 0 Ψ(r)Ψ(r ) = δ(r r ) 0 d d x exp ( x2 4t ) φ(x, 0) dr r 1 2 (d 1) e r 2 4t Ψ(r) 1 d d x φ(x, 0) r r< x <r+ r Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

65 A heuristic argument Diffusion equation φ(x = 0, t) L 2 Ψ(u) Ψ(u ) = δ(u u ) 0 du u d 2 4 e u t Ψ(u) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39

66 A heuristic argument Diffusion equation φ(x = 0, t) L 2 Ψ(u) Ψ(u ) = δ(u u ) 0 du u d 2 4 e u t Ψ(u) Random polynomials : K n (x) = a 0 + n i=1 a ii d 2 4 x i K n (1 1/t) a 0 + n n i=1 a(u)a(u ) = δ(u u ) 0 i d 2 4 e i t a i du u d 2 4 e u t a(u) Grégory Schehr (LPTMS Orsay) Rand. polynomials & the heat equation Darmstadt, July / 39