“Integrable” gap probabilities for the Generalized Bessel process “Integrable” gap probabilities for the Generalized Bessel process Manuela Girotti SISSA, Trieste, June 7th 2017 1 / 27
“Integrable” gap probabilities for the Generalized Bessel process Table of contents 1 Introduction: the Generalized Bessel process 2 First result: differential identity for gap probabilities Sketch of the proof 3 Painlev´ e and hamiltonian connection (joint with M. Cafasso, U. Angers) Painlev´ e-type equation Garnier system 4 Conclusive remarks and open questions 2 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process 1 Introduction: the Generalized Bessel process 2 First result: differential identity for gap probabilities Sketch of the proof 3 Painlev´ e and hamiltonian connection (joint with M. Cafasso, U. Angers) Painlev´ e-type equation Garnier system 4 Conclusive remarks and open questions 3 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process BESQ α model Consider a system of n independent squared Bessel paths BESQ α { X 1 ( t ) , . . . , X n ( t ) } with parameter α > − 1, conditioned never to collide. The process { � X ( t ) } t ≥ 0 is a diffusion process on [0 , + ∞ ) n . Additionally, we impose initial and final conditions X j (0) = a > 0 and X j ( T ) = 0 ∀ j = 1 , . . . , n. 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process BESQ α model Consider a system of n independent squared Bessel paths BESQ α { X 1 ( t ) , . . . , X n ( t ) } with parameter α > − 1, conditioned never to collide. The process { � X ( t ) } t ≥ 0 is a diffusion process on [0 , + ∞ ) n . Additionally, we impose initial and final conditions X j (0) = a > 0 and X j ( T ) = 0 ∀ j = 1 , . . . , n. 4 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process BESQ α model Consider a system of n independent squared Bessel paths BESQ α { X 1 ( t ) , . . . , X n ( t ) } with parameter α > − 1, conditioned never to collide. The process { � X ( t ) } t ≥ 0 is a diffusion process on [0 , + ∞ ) n . Additionally, we impose initial and final conditions X j (0) = a > 0 and X j ( T ) = 0 ∀ j = 1 , . . . , n. 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process The joint probability density is given as � n xj 1 � n � � e − x j − 1 p α +1 − j (mod 2) x k − 1 det ( a, x k ) j,k =1 det 2( T − t ) d x 1 . . . d x n t k j Z n,t j,k =1 = 1 n ! det [ K n ( x i , x j ; t )] n i,j =1 d x 1 . . . d x n � √ xy � y � α/ 2 e − x + y � 1 where p α t ( x, y ) is the transition probability p α 2 t I α t ( x, y ) = 2 t x t and the correlation kernel K n given in terms of MOP with weights depending on the Bessel functions I α . Remark (Random Matrix interpretation) Let M ( t ) be a p × n matrix with independent complex Brownian entries (with mean zero and variance 2 t ). The set of singular values { λ 1 ( t ) , . . . , λ n ( t ) } , λ i ( t ) ≥ 0 ∀ i i.e. the eigenvalues of the product M ( t ) ∗ M ( t ) , has the same distribution as the above noncolliding particle system BESQ α with α = 2( n − p + 1) (K¨ onig, O’Connell, ’01). 5 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process (Double) Scaling limit Starting from the kernel K n , one can perform a double scaling limit as n ր + ∞ in different parts of the domain of the spectrum: the sine kernel appears in the bulk, the Airy kernel at the soft edges and the Bessel kernel appears at the hard edge x = 0 (Kuijlaars et al. , ’09). At a critical time t ∗ , there is a transition between the soft and the hard edges and the local dynamics is described by a new critical kernel. 6 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process The Generalized Bessel kernel Theorem (Kuijlaars, Martinez-Finkelshtein, Wielonsky, ’11) � c ∗ x c ∗ n 3 / 2 , c ∗ y n 3 / 2 ; t ∗ − c ∗ τ � = K crit lim n 3 / 2 K n √ n ( x, y ; τ ) x, y ∈ R + , τ ∈ R , α n ր + ∞ with e xu + τ 2 u 2 − yv − τ 1 1 u + � u v − d u d v 2 v 2 � � � α K crit ( x, y ; τ ) = . α 2 πi 2 πi v − u v Γ Σ 7 / 27
“Integrable” gap probabilities for the Generalized Bessel process Introduction: the Generalized Bessel process Gap probabilities of the Generalized Bessel process Our object of study are the gap probabilities , meaning the probability of finding no points in a given domain. For a determinantal process with kernel K n , this boils down to calculating a Fredholm determinant: ∞ ( − 1) k � � P ( X min > s ) = 1 + [0 ,s ] k det [ K n ( x i , x j )] i,j =1 ,...,k d x 1 . . . d x k k ! k =1 � � � � = det Id L 2 ( R + ) − K n � � [0 ,s ] and in the scaling limit regime � � � � → det Id L 2 ( R + ) − K crit � � det Id L 2 ( R + ) − K n as n ր + ∞ . � α � � � � 0 , c ∗ s � [0 ,s ] n 3 / 2 8 / 27
“Integrable” gap probabilities for the Generalized Bessel process First result: differential identity for gap probabilities 1 Introduction: the Generalized Bessel process 2 First result: differential identity for gap probabilities Sketch of the proof 3 Painlev´ e and hamiltonian connection (joint with M. Cafasso, U. Angers) Painlev´ e-type equation Garnier system 4 Conclusive remarks and open questions 9 / 27
“Integrable” gap probabilities for the Generalized Bessel process First result: differential identity for gap probabilities Differential identity Theorem (Girotti, ’14) Let s > 0 and K crit be the integral operator acting on L 2 ( R + ) with kernel defined α above. Then, the following differential formula for gap probabilites holds � � � � � Y − 1 ˆ ˆ Id L 2 ( R + ) − K crit � d s,τ ln det = ( Y 1 ) 2 , 2 d s − 2 , 2 d τ Y 1 α � 0 � [0 ,s ] where Y is the solution to a suitable RH problem and Y 1 and ˆ Y j are the coefficients appearing in the asymptotic expansion of Y at infinity and in a neighbourhood of zero, respectively. 10 / 27
“Integrable” gap probabilities for the Generalized Bessel process First result: differential identity for gap probabilities The Riemann-Hilbert problem for Y Find a 2 × 2 matrix-valued function Y = Y ( λ ; s, τ ) such that Y is analytic on C \ (Γ ∪ Σ) Y admits a limit when approaching the contours from the left Y + or from the right Y − (according to their orientation), and the following jump condition holds − λ − α e − λs − τ 1 � � λ − 1 2 λ 2 λ ∈ Σ 0 1 Y + ( λ ) = Y − ( λ ) � � 1 0 λ ∈ Γ − λ α e λs + τ 1 λ + 2 λ 2 1 Y has the following (normalized) behaviour at ∞ : � 1 Y ( λ ) = I + Y 1 ( s, τ ) � + O λ → ∞ . λ 2 λ 11 / 27
“Integrable” gap probabilities for the Generalized Bessel process First result: differential identity for gap probabilities Sketch of the proof Sketch of the proof Proposition The following identity holds � � � � � Id L 2 ( R + ) − K crit � det = det Id L 2 (Σ ∪ Γ) − H α � � [0 ,s ] where H is an Its-Izergin-Korepin-Slavnov (’90) integral operator with kernel H = f ( λ ) T g ( µ ) λ − µ µs + τ 1 µ + � � 2 µ 2 χ Γ ( µ ) e − λs 1 µ α e 2 χ Σ ( λ ) . f ( λ ) = g ( µ ) = − µs 2 − τ 1 µ − 2 πi χ Γ ( λ ) 2 µ 2 χ Σ ( µ ) µ − α e � The result can be proved by noticing that K crit � is unitarily equivalent (via α � � [0 ,s ] Fourier transform) to a certain integral operator that can be decomposed as the above operator H . 12 / 27
“Integrable” gap probabilities for the Generalized Bessel process First result: differential identity for gap probabilities Sketch of the proof IIKS operators naturally carry an associated RH problem, whose solution Y is tied to the invertibility of their resolvent operator. Given such RH problem, we make use of a major (and more general) result due to Bertola (’10) and Bertola-Cafasso (’11) which, if applied to our case, reads as follows Theorem (Bertola-Cafasso, ’11) Define the quantity for ρ = s, τ − ( ∂ ρ J ) J − 1 � d λ � � Y − 1 − Y ′ ω ( ∂ ρ ) := Tr 2 πi . Σ ∪ Γ Then, we have the equality � � ω ( ∂ ρ ) = ∂ ρ ln det Id L 2 (Σ ∪ Γ) − H . By expanding the solution Y at infinity and at zero, this identity can be further simplified and explicitly calculated and it yields the final result: � � � � Y − 1 ˆ ˆ d s,τ ln det Id L 2 (Σ ∪ Γ) − H = ( Y 1 ) 2 , 2 d s − Y 1 2 , 2 d τ. 0 13 / 27
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