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Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach

We consider the gap probability for the Generalized Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of Its-Izergin-Korepin-Slavnov integrable kernels and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism, while in the multi-time case we explicitly define a new multi-time kernel to study.


Introduction
The Generalized Bessel process is a determinantal point process [11] defined in terms of a trace-class integral operator acting on L 2 (R + ), with kernel with ν > −1; the logarithmic cut is on R − . The curve γ andγ are described in Figure 2. The Generalized Bessel kernel was first introduced as a critical kernel by Kuijlaars et al. in [8] and [9]. In these articles, a model of non-intersecting squared-Bessel paths, starting at time t = 0 at the same positive value x = κ > 0 and ending at time t = 1 at x = 0, was proposed and studied.
The positions of the paths at any given time t ∈ (0, 1) are a determinantal point process with correlation kernel built out of the transition probability density of the squared Bessel process. In [8], it was proven that, after appropriate scaling, the paths fill out a region in the tx-plane as in Figure  1: the paths stay initially away from the axis x = 0, but at a certain critical time t * the smallest paths come to the hard edge x = 0 and then remain close to it. Figure 1: Numerical simulation of 50 rescaled non-intersecting Squared Bessel Paths with κ = 1 (taken from [8]).
As the number of paths tends to infinity, the local scaling limits of the correlation kernel are the following: the sine kernel appears in the bulk, the Airy kernel at the soft edges, i.e. the upper boundary for all t ∈ (0, 1) and the lower boundary of the limiting domain for t < t * , while for t > t * , the Bessel kernel appears at the hard edge x = 0, see [8,.
Thus, in the critical time t = t * , there is a transition between the Airy and the Bessel kernel and the dynamics at that point is described by the new (critical) kernel (1).
We prefer to refer to this kernel, and to its multi-time counterpart, as "Generalized Bessel kernel" because of the several analogies with the Bessel kernel ( [5]) appearing along our study. First of all, the contours setting is the same one as for the Bessel kernel (see [5]); many of the calculations performed in [5] for the Bessel kernel are here reproduced with very few adjustments. Moreover, as it will be clear in Section 2, gap probabilities of the Generalized Bessel operator are related to a Lax pair which belongs to a higher order Painlevé III hierarchy, while it is known that the gap probabilities of the Bessel kernel is related to the Painlevé III transcendent, as shown in [5].
In this paper we will show that the correlation functions related to the Generalized Bessel process can be expressed in terms of determinants of a kernel K(x, y) (matrix-valued in the multi-time case) in the sense of Its-Izergin-Korepin-Slavnov ( [7]). Namely, the kernel can be written in the form where f, g are two matrices of a given dimension such that f T (x)· g(x) = 0. We refer to the review paper [11] that also explains how the "gap probabilities" (probability of having no particles in certain regions) are related to Fredholm determinants. In such case, the computation of the Fredholm determinant is reduced to the study of a certain Riemann-Hilbert problem canonically related to the operator's kernel (for a concise account, see [6]). The Riemann-Hilbert formulation is very useful for finding some differential equations and studying asymptotic properties of the determinant. Moreover, it will be possible to connect it to the Jimbo-Miwa-Ueno τ function.
Our approach is the same as the one used in [3] for the scalar Airy and Pearcey operators, in [2] for the matrix Airy and Pearcey operators and in [5] for the Bessel operator.
As an example of possible applications we describe how to obtain a system of isomonodromic Lax equations for the (single-time) process.
Moreover, having a RiemannHilbert formulation for these Fredholm determinants will allow the study of asymptotics of Generalized Bessel gap probabilities and their connection with Airy and Bessel gap probabilities, using steepest descent methods, along the lines of [3].
The formulation of the multi-time Generalized Bessel kernel is a completely new result and its derivation has been addressed in the Appendix. An equivalent formulation has been proposed and autonomously derived by Delvaux and Veto ([12]).
The paper is organized as follows: in section 2 we will deal with the single-time Generalized Bessel operator restricted to a generic collection of intervals; in the subsection 2.2 we will focus on the single-time Generalized Bessel process restricted to a single interval [0, a]: we will find a Lax pair and we will be able to make a connection between the Fredholm determinant and a Painlevé III hierarchy. In section 3 we will study the gap probabilities for the multi-time Bessel process. In the Appendix, we show how we found the multi-time Generalized Bessel kernel and we make a comparison with Delvaux and Veto's one. We prove that these two kernels are equivalent up to a transposition of the operator and a translation of the parameter τ .

Single-time Generalized Bessel
The Generalized Bessel kernel is where τ ∈ R is a fixed parameter, the contourγ is a closed loop in the right half-plane tangent to the origin and oriented clockwise, while the contour γ is an unbounded loop oriented counterclockwise and encirclingγ; the logarithmic cut lies on R − (see Figure 2). Remark 1. The curve setting is equivalent to the curve setting appearing in the definition of the Bessel kernel (see [5]).
Moreover, the phase appearing in the exponential (4) resembles the Bessel kernel one ψ(z, t) := zt − 1 4t with an extra term which introduces a higher singularity at 0.
Our interest is focused on the gap probability of such operator restricted to a collection of intervals I, i.e. the quantity with χ I the characteristic function of the Borel set I.
Remark 3. The Generalized Bessel operator is not trace class at infinity; indeed, the kernel K GEN (x, x; τ ) is not integrable in a neighbourhood of ∞.
The first step in our study is to establish a relation between the Generalized Bessel operator and a suitable integrable operator in the sense of Its-Izergin-Korepin-Slavnov (IIKS; see [7]).
Theorem 4. Given a collection of (disjoint) intervals I := N k=1 [a 2k−1 , a 2k ], the following identity between Fredholm determinats holds where K GEN is an IIKS integrable operator acting on L 2 (γ ∪γ) with kernel Proof. Since the preliminary calculations are linear, we will start working on the single term K GEN a j and we will later sum them up over the j = 1, . . . , 2N .
where we continuously deformed the contour γ into a suitably translated imaginary axis. Introducing the following Fourier transform operators we can claim that being an operator acting on L 2 (iR+ ) with kernels ∀ j = 1, . . . , 2N , ξ, t ∈ iR + . In order to ensure the convergence of the kernel, we conjugate it with the function f (z) := e a 1 z 2 and, with abuse of notation, we call the resulting kernels K GEN and K GEN a j as well.
Remark 5. We recall that Fredholm determinants are invariant under conjugation by bounded invertible operators.
We continuously deform the translated imaginary axis iR + into its original shape γ; note that a j − a 1 2 > 0, ∀j = 1, . . . , 2N . It can be easily shown that the operator K GEN a j is the composition of two operators for every j = 1, . . . , 2N ; moreover, it is trace-class. Lemma 6. The operators K GEN a j are trace-class operators, ∀ j = 1, . . . , 2N , and the following decomposition holds K GEN (16) A j,1 and B 1 are trace-class operators themselves.
Proof. We introduce an additional translated imaginary axis iR + δ (δ > 0), not intersecting with γ andγ, and we decompose A j,1 and B 1 in the following way: and All the kernels involved are of the form K(z, w) with z and w on two disjoint curves, say C 1 and C 2 . It is sufficient to check that C 1 ×C 2 |K(z, w)| 2 |dz||dw| < ∞ to ensure that the operator belongs to the class of Hilbert-Schmidt operators. This implies that {A j,1 } j , B 1 and K GEN a j are trace-class (for all j = 1, . . . , 2N ), since composition of two HS operators. Now we recall that any operator acting on a Hilbert space of the type H = H 1 ⊕ H 2 can be decomposed as a 2 × 2 matrix of operators with (i, j)entry given by an operator H j → H i . Thus, we can perform a chain of equalities the second equality follows from the multiplication on the left by the matrix (with determinant = 1) and the operator K GEN is an integrable operator with kernel as in the statement.

Riemann-Hilbert problem and τ -function
We can proceed now with building a Riemann-Hilbert problem associated to the integrable kernel we just found in Theorem 4. This will allow us to find some explicit identities for its Fredholm determinant.
Proposition 7. Given the integrable kernel (8)-(10), the correspondent RHproblem is the following: finding an (2N + 1) × (2N + 1) matrix Γ such that it is analytic on C\Ξ (Ξ := γ ∪γ) and Proof. We simply need to verify that It is easy to see that the jump matrix is conjugate to a matrix with with a the collection of all endpoints {a j }. Thus, considering the matrix Ψ(λ, a) := Γ(λ, a)e T (λ, a) , Ψ satisfies a RHproblem with constant jumps, thus it's (sectionally) a solution to a polynomial ODE.
Referring on the results stated in [1] and [3] and adapted to the case at hand, we can claim that Theorem 8. For every parameter ρ, on which the Generalized Bessel operator may depend, Moreover, thanks to the Jimbo-Miwa-Ueno residue formula (see [3]), As far as the parameter τ is concerned, the following result holds whereΓ 0 andΓ 1 are coefficients appearing in the asymptotic expansion of the matrix Γ in a neighbourhood of zero.
Proof. The phases θ a j are linear in a j , exactly as in the Bessel kernel case (see [5]).

The single-interval case
In case we consider a single interval I = [0, a], we are able to perform a deeper analysis on the gap probability of the Generalized Bessel operator and link it to an explicit Lax pair.
We will see that the Lax pair A and U will recall the Bessel Lax pair very closely (see [5]), except for the presence of an extra term for the spectral matrix A. Such term will introduce a higher order Poincaré rank at λ = 0 as it will be clear in the following calculations. Moreover, thanks to the presence of the parameter τ other than the endpoint a, we can actually calculate a Lax "triplet".
First of all, we reformulate Theorems 4 and 8, focusing on our present case.
Theorem 10. Given I = [0, a], the following equality between Fredholm determinants holds The associated RH-problem reads as follows: with Γ a 2 × 2 matrix, analytic on analytic on the complex plane except on the collection of curves Ξ, along which the above jump condition is satisfied with jump matrix M (λ) : Thus the jump matrix M is equivalent to a matrix with constant entries, via the conjugation e Ta(λ) , T a (λ) = 1 2 θ a σ 3 , where θ a := −λa − τ λ + 1 2λ 2 + ν ln λ and σ 3 is the third Pauli matrix. This allows us to define the matrix Ψ(λ) := Γ(λ)e Ta(λ) which solves a RHP with constant jumps and is (sectionally) a solution to a polynomial ODE: Applying Theorem 8, we get Theorem 11.
for every parameter ρ on which the operator K GEN depends.
In particular, thanks to the Jimbo-Miwa-Ueno residue formula, we have with Γ 1;2,2 the (2, 2)-entry of the residue matrix Γ 1 at ∞, while theΓ j 's appear in the asymptotic expansion of Γ near zero.
We can now calculate the Lax "triplet" associated to the RH-problem above: with coefficients We point out that λ = 0 is an irregular point of Poincaré rank 2. The behaviour at zero shows a higher order rank with respect to the Lax pair for the Bessel operator [5], where the point λ = 0 was of rank 1.
Moreover, the matrix U is the same as the one appearing in the Bessel Lax pair (in the non-rescaled case, see [5]).
The expression of the Lax pair A and U suggests that their compatibility equation will eventually lead to a higher order ODE belonging to the Painlevé III hierarchy.

Multi-time Generalized Bessel
The multi-time Generalized Bessel operator on L 2 (R + ) with times τ 1 < . . . < τ n is defined through a matrix kernel with entries the curve γ is the same one as in the single-time Extended Bessel kernel (a contour that winds around zero counterclockwise an extends to −∞) and γ := 1 γ ; ∆ ji := τ j − τ i > 0. Remark 13. The matrix P ∆ ij is strictly upper triangular.
Remark 14. The above definition of the multi-time kernel is the one given by Delvaux and Veto ([12]). We preferred to use this one because the study of the gap probability with this expression involves less complicated calculations than with our equivalent version (see Appendix).
As in the single-time case, we are again interested in the gap probability of the operator restricted to a collection intervals I j at each time τ j (∀ j), i.e. det where χ I = diag (χ I 1 , . . . , χ In ) is a diagonal matrix of characteristic functions and 2k j ] ∀ j = 1, . . . , n.
Remark 15. The multi-time Bessel operator fails to be trace-class on infinite intervals.
For the sake of clarity, we will focus on the simple case I j = [0, a (j) ], ∀ j. The general case follows the same guidelines described below; the only difficulties are mostly technical, due to heavy notation, and not theoretical.
As in the single-time case, we start by establishing a link between the multi-time Generalized Bessel operator and a suitable IIKS operator, which we will examine deeper in the next subsection.
Theorem 16. The following identity between Fredholm determinants holds with χ I = diag (χ I 1 , . . . , χ In ) the characteristic matrix of the collection of intervals. The operator K B is an integrable operator acting on the Hilbert space with γ −k := 1 γ − 4τ k . Its kernel is a 2n × 2n matrix of the form where f, g are N × 2n matrices, with N = 2n + (n − 1) = 3n − 1.
Remark 17. By Fredholm determinant " det " we denote the determinant defined through the usual series expansion with K an integral operator acting on the Hilbert space L 2 (X, dµ(x)) and kernel K(x, y).
In the case at hand, we will see that the operator K GEN χ I = (H +P ∆ )χ I is the sum of a trace-class operator (Hχ I ) plus a Hilbert-Schmidt operator (P ∆ χ I ) with diagonal-free kernel.
Therefore the naming of Fredholm determinant refers to the following expression: where det 2 denotes the regularized Carleman determinant (see [10]).
Proof. Thanks to the invariance of the Fredholm determinant under kernel conjugation, we can discard the term y x ν in our further calculations.
We will work on the entry (i, j) of the kernel. We can notice that for x < 0 or y < 0 the kernel is identically zero, K GEN (x, y) ≡ 0. Then, applying Cauchy's theorem, we have where we deformed γ into a translated imaginary axis iR + ( > 0) in order to make Fourier operator defined below more explicit; the last equality follows from the change of variable on s = 1/(v + 4τ j ), thus the contourγ becomes similar to γ and can be continuously deformed into it.
On the other hand, It is easily recognizable the conjugation with a Fourier-like operator as in (12), so that Now we can perform the following change of variables on the Fouriertransformed kernel so that the kernel will have the final expression The obtained (Fouriertransformed) Generalized Bessel operator is an operator acting on L 2 ( n k=1 γ −k , C n ) ∼ n k=1 L 2 (γ −k , C n ). Lemma 18. The following decomposition holds K GEN = M • N + P, with M, N , P Hilbert-Schmidt operators with kernel entries Proof. As in Lemma 6, all the kernels involved are of the form K(z, w) with z and w on two disjoint curves, say C 1 and C 2 . The Hilbert-Schmidt property it thus ensured by simply checking that C 1 ×C 2 |K(z, w)| 2 |dz||dw| < ∞.
We define the Hilbert space and the matrix operator K GEN : H → H For now, we denote by " det " the determinant defined by the Fredholm expansion (66); then, " det "(Id − K B ) = det 2 (Id − K B ), since its kernel is diagonal-free. We also introduce another Hilbert-Schmidt operator whose Carleman determinant (det 2 ) is still well defined and det 2 (I−K GEN,2 ) is identically 1. We finally perform the following chain of equalities The first equality follows from the fact that K GEN − H is diagonal-free; the second equality follows from invariance of the determinant under Fourier transform; the third identity is an application of the following result: given A, B Hilbert-Schmidt operators, then It is finally just a matter of computation to show that K B is an integrable operator of the form (57)-(65).

Rieman-Hilbert problem and τ -function
We can now relate the Fredholm determinant of the multi-time Generalized Bessel operator to the isomonodromy theory by defining a suitable Riemann-Hilbert problem.
2 × 2 case. In the simple 2-times case, the jump matrix reads The jump matrix, though it might look complicated, is equivalent to a matrix with constant entries so that the matrix Ψ GEN (λ) = Γ(λ)e T GEN (λ) solves a Riemann-Hilbert problem with constant jumps and it is a solution to a polynomial ODE.
Proof. The following formula holds in general (see [1]) where Therefore, it is enough to verify that H(M ) ≡ 0 with M (λ) = I − J GEN (λ).
Moreover, making use of the Jimbo-Miwa-ueno residue formula (see [1]), it can be shown that Theorem 21. The following equality holds In particular, regarding the derivative with respect to the endpoints a (j) (j = 1, . . . , n) Regarding the derivative with respect to τ Finally, regarding the derivative with respect to the times τ j (j = 1, . . . , n) where, given the asymptotic expansion of the matrix Γ ∼ Γ 0 + λ k Γ 1 + λ 2 k Γ 2 + · · · in a neighbourhood of −4τ k , we defined Φ 0 := Γ −1 0 Γ 1 and Φ 1 : Remark 22. We stated the second part of the theorem above in the simple case I = diag χ [0,a (1) ] , . . . , χ [0,a n ] in order to avoid heavy notation. The general case follows the same guidelines shown in the proof.
Proof. We will calculate the residues separately and we will focus on the different parameters (a (j) , τ j and τ ). Residue at ∞. There's no contribution from the residue at infinity when we consider the derivative with respect to the endpoints a (j) . On the other hand, taking the derivative with respect to the times τ k gives: thus the residue is We follow a similar argument for the parameter τ : Thus, Residue at 4τ k . We recall the asymptotic expansion of the matrix Γ in a neighbourhood of −4τ k : Remark 23. Note that the asymptotic expansion near −4τ k is, in general, different for each k, but we wrote them in this way in order to avoid heavy notation.
Regarding the derivative with respect to the endpoints a (k) , we have which implies and regarding the derivative with respect to the times τ k , we have thus, There is no contribution from the residue at −4τ k (k = 1, . . . , n) when taking the derivative with respect to τ .

A Building the multi-time Generalized Bessel kernel
The starting point of this investigation is the known single-time kernel discovered by Kuijlaars et al. ([9]) where the curve γ is an unbounded curve that extends from −∞ to zero and then back to −∞, encircling the origin in a counterclockwise way, and γ := 1 γ ; the logarithmic cut is on R − , as shown in Figure 2. The diffusion kernel related to the Squared Bessel Paths is where ∆ > 0 represents the gap between two given times τ i and τ j and I ν is the modified Bessel function of first kind (the same diffusion kernel appears in the definition of the multi-time Bessel kernel; see [5]). The extended multi-time kernel is given by where in particular P ∆ is a strictly upper-triangular matrix with (i, j)-entry P ∆ ij := p(x, y, ∆ ij ) when j > i (∆ ij := |τ i − τ j | > 0). This is essentially the derivation in [4] applied to case at hand.
The proof is based on the verification that the definition of the kernel above satisfies the Eynard-Mehta theorem on multi-time kernels ([4]).
Independently from the present work and almost simultaneously, Veto and Delvaux ( [12]) introduced another version of the multi-time Generalized Bessel operator, called Hard-edge Pearcey process.
The kernel of the Hard-edge Pearcey reads L ν := H − P ∆ with entries and P ∆ the usual transition density as above. Γ −τ i is a clockwise oriented closed loop which intersects the real line at a point to the right of −τ i , and also at −τ i itself, where it has a cusp at angle π; δ > 0 is chosen such that the contour iR + δ passes to the right of the singularity at −t and to the right of the contour Γ −τ i . The logarithmic branch is cut along the negative half-line.
Proposition 26. The Hard-edge Pearcey operator is the transposed of the Generalized Bessel operator (125)-(126) defined above. More precisely, Proof. The results comes from straightforward changes of variables.
Corollary 27 ( [12]). In the single-time case (τ i = τ j ), both the Generalized Bessel kernel and the Hard-edge Pearcey kernel coincide with the single-time kernel defined in [9], up to a transposition: L ν (x, y; σ)| ∆ ij =0 = y x ν K KM W ν (y, x; τ i +σ) = y x ν K GEN (y, x; τ i + σ) ∆ ij =0 (138) Remark 28. Since the Fredholm determinant is invariant under transposition, we preferred to work on the version given by Veto and Delvaux for the multi-time Generalized Bessel operator, because of more straighforward calculations which reminds more closely the ones performed for the Bessel process ( [5]).