Abstract
An explicit expression for the expected value of a regularized multiplicative functional under the sine-process is obtained by passing to the scaling limit in the Borodin–Okounkov–Geronimo–Case formula.
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1. Introduction
1.1. Statement of the Main Result
The sine-process is the scaling limit of radial parts of Haar measures on the unitary groups of growing dimension. Expectations of multiplicative functionals with respect to the sine-process admit an explicit expression for observables of Sobolev regularity \(1/2\) with bounded Hilbert transform. The explicit expression is obtained as the scaling limit of the Borodin–Okounkov–Geronimo–Case formula for Toeplitz determinants. Recall that the Borodin–Okounkov–Geronimo–Case formula gives the remainder term in Ibragimov’s version of the Strong Szegő Theorem.
Recall that the sine-process, denoted by \(\mathbb P_{\mathscr{S}}\), is a determinantal point process with the sine kernel
which is the kernel of the projection operator on the Paley–Wiener space
In other words, the sine-process is a measure on the space of configurations \(\operatorname{Conf}(\mathbb{R})\), that is, the space of subsets \(X\subset\mathbb{R}\) without accumulation points. The sine-process is uniquely defined by the condition
valid for any bounded Borel function \(f\) with compact support. Theorem 1 below gives a convenient expression for the expectation (1) for \(1/2\)-Sobolev regular functions \(f\) with bounded Hilbert transform.
To any Borel bounded function \(f\) with compact support, we assign an additive functional \(S_f\) on \(\operatorname{Conf}(\mathbb{R})\) by the formula
the series on the right-hand side contains only a finite number of non-zero terms. The variance of the additive functional \(S_f\) is given by the formula
Following [7], we define the space \(\dot H_{1/2}(\mathscr{S})\) as the completion of the family of compactly supported smooth functions on \(\mathbb{R}\) with respect to the norm \({\|\cdot\|}_{\dot H_{1/2}(\mathscr{S})}\) given by the formula
By definition, the correspondence \(f\mapsto S_f-\mathbb{E}_{\mathbb P_{\mathscr{S}}}S_f\) is extended by continuity onto the entire space \(\dot H_{1/2}(\mathscr{S})\). Let us denote
and refer to \(\overline{S}_f\) as the regularized additive functional for the function \(f\in\dot H_{1/2}(\mathscr{S})\).
Below we give an explicit formula for the exponential moments of regularized additive functionals with \(1/2\)-Sobolev regular functions \(f\) for the sine-process. Let us recall some basic definitions.
We use the following convention for the Fourier transform on the real line:
Denote by the symbol \(\widetilde{\phantom{a}}\) the reflection with respect to zero:
Define the space of Sobolev type \(\dot H_{1/2}(\mathbb{R})\) as the completion of the family of smooth compactly supported functions with respect to the norm
By the symbol \(\langle{\,\cdot\,},{\,\cdot\,}\rangle_{\dot H_{1/2}(\mathbb{R})}\), we denote the bilinear form given by the formula
Therefore,
Further, for a function \(f\in\dot H_{1/2}(\mathbb{R})\), let \(f_+\), \(f_-\) be the functions defined by the formulae
Finally, let
Clearly, \(f_--f_+\) is the Hilbert transform of the function \(f\) multiplied by \(\sqrt{-1}\).
For a function \(r\in L_2(\mathbb{R})\cap L_\infty(\mathbb{R})\), we denote by \(\mathfrak{H}(r)\) the continual Hankel operator acting by the formula
Denote by the symbol \(\mathscr{H}(1/2,\infty)\) the completion of the space of smooth compactly supported functions on \(\mathbb{R}\) with respect to the norm
One can see from the definition that for \(h\in\mathscr{H}(1/2,\infty)\), the operator \(\mathfrak{H}(h)\) is Hilbert–Schmidt.
Further, \(f_1,f_2\in\mathscr{H}(1/2,\infty)\) implies \(f_1f_2\in\mathscr{H}(1/2,\infty)\). Therefore, \(f\in\mathscr{H}(1/2,\infty)\) holds if \(\exp(f)-1\in\mathscr{H}(1/2,\infty)\).
The expectation of a multiplicative functional of the sine-process corresponding to a \(1/2\)-Sobolev regular function is given as follows.
Theorem 1.
Let \(f\in \dot H_{1/2}(\mathbb{R})\) satisfy \(f_--f_+\in L_\infty(\mathbb{R})\). Then
The condition \(f_--f_+\in L_\infty(\mathbb{R})\) immediately implies
The boundedness condition may be omitted if \(f\) is real-valued.
Corollary 1.
Formula (3) holds if \(f\in\dot H_{1/2}(\mathbb{R})\) is real-valued.
Proof.
If \(f\) is real-valued, then \(|h|\equiv 1\), \(h\in\mathscr{H}(1/2,\infty)\), and \(\|\mathfrak{H}(h)\|\le 1\), so formula (3) holds even if \(f\) is unbounded. \(\square\)
Theorem 1 is proven by passing to a scaling limit in its discrete counterpart, the Borodin–Okounkov–Geronimo–Case formula.
It is enough to prove Theorem 1 for smooth functions with compact support. The general case immediately follows from the continuity of both sides in (3) in the space \(\mathscr{H}(1/2,\infty)\).
Remark 1.
Under more restrictive assumptions on \(f\), an equivalent of formula (3) was obtained by Basor and Chen [2]. A special case of formula (3) was used in [8] to prove that almost all realizations of the sine-process have excess one in the Paley–Wiener space.
1.2. Historical Remarks
Gabor Szegő proved the Polya conjecture, which is now known as the First Szegő Theorem [15], in 1915, the same year he enlisted in the Royal Honvéd cavalry as a volunteer. The Second (or Strong) Szegő Theorem [16] was proven 37 years later in 1952. For a review of its history, see [9], [14]. Necessary and sufficient conditions under which the theorem holds were derived by Ibragimov [13], [11]. Geronimo and Case [10] proved formula (5) in 1979. It was later rediscovered in 2000 by Borodin and Okounkov [4]. Now there exist several proofs of the Borodin–Okounkov–Geronimo–Case formula [3], [5], [6], but the question concerning necessary and sufficient conditions for the formula is still open.
2. Beginning of the Proof of Theorem 1: Borodin–Okounkov–Geronimo–Case Formula
Let us recall the Borodin–Okounkov–Geronimo–Case formula.
Set \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\). For a function \(F\in L_1(\mathbb{T})\) with Fourier expansion
we define the Toeplitz operator \(T(F)\) with symbol \(F\) as the operator acting on functions \(\varphi\) with finite support on \(\mathbb{N}\) by the formula
The Hankel operator \(\mathbf{H}(F)\) with the symbol \(F\) is defined by the formula
If \(F\in L_\infty(\mathbb{T})\), operators \(T(F)\) and \(\mathbf{H}(F)\) are bounded on \(l_2(\mathbb{N})\). As in the case of the real line, we denote the reflection with respect to zero by the symbol \(\widetilde{\phantom{a}}\):
Let \(D_n(F)\) stand for the \(n\times n\) Toeplitz determinant corresponding to a symbol \(F\), that is,
The Andreief formula [1]
implies that for any function \(G\in L_1(\mathbb{T})\), we have
where
is the \(n\)-th Dirichlet kernel.
As in § 1, we introduce the Sobolev space of order \(1/2\) on the unit circle \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\), the respective seminorm, and the bilinear form, respectively, as follows:
Therefore,
Following Borodin and Okounkov, we consider a square-integrable function on the unit circle \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\)
with zero average: \(\widehat F(0)=0\). Let
Define the function \(N\) by the formula
We are now ready to formulate the Borodin–Okounkov–Geronimo–Case theorem.
Theorem 2 (see [4], [10]).
Let \(F\in H_{1/2}(\mathbb{T})\), \(\widehat F(0)=0\), and \(F_--F_+\in L_\infty(\mathbb{T})\). Then we have
If \(F\) is real-valued, then \(|N(\theta)|=|\widetilde{N^{-1}}(\theta)|=1\) for all \(\theta\in\mathbb{T}\).
It is clear from the definitions that the Hankel operators \(\mathbf{H}(N)\) and \(\mathbf{H}(\widetilde{N^{-1}})\) are adjoint to each other.
Corollary 2.
If \(F\in H_{1/2}(\mathbb{T})\) is real-valued, then (5) holds.
Borodin and Okounkov emphasize that identity (5) can be considered as an equality between formal power series in \(\widehat{F}(k)\).
Remark 2.
The Second Szegő theorem for \(G=\exp(F)\) states that
holds under the assumptions that \(F\in H_{1/2}(\mathbb{T})\) and \(\widehat{F}(0)=0\). If the function \(F\) is not assumed to be real-valued, the additional condition \(F_--F_+\in L_\infty(\mathbb{T})\) is required in the proof of the Borodin–Okounkov–Geronimo–Case formula to ensure that the respective Hankel operators are Hilbert–Schmidt.
Question.
Is the condition \(F_--F_+\in L_\infty(\mathbb{T})\) necessary for the Borodin–Okounkov–Geronimo–Case formula to hold? What are the necessary and sufficient conditions?
Remark 3.
Our definition differs slightly from the conventions of Borodin and Okounkov, who considered the function \(F_-(\pi-\theta)-F_+(\pi-\theta)\). However, the resulting kernels differ by a gauge factor of \((-1)^{i+j}\) and, therefore, have the same Fredholm determinants.
3. Scaling Limit of the Borodin–Okounkov–Geronimo–Case Formula
Our next step is to pass to the scaling limit in the Borodin–Okounkov–Geronimo–Case formula under the scaling \(R_n(\theta)=r(n\theta)\).
Let \(r\) be a smooth compactly supported function with zero average. For large enough \(n\in\mathbb{N}\), the support of the function \(r(n\varphi)\) lies in the interval \((-\pi, \pi)\). Define the functions \(R_n\) on \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\) by the formula
We now introduce the following decompositions into the positive and negative harmonics: \(r=r^+-r^-\), \(R_n=R_n^++R_n^-\), and
Here and below we denote Fourier transforms on \(\mathbb{R}\) and \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\) by \(\widehat f^{\,\,\mathbb{R}}\) and \(\widehat f^{\,\,\mathbb{T}}\), respectively. By definition, we have
Lemma.
Let \(r\) be a smooth compactly supported function on \(\mathbb{R}\) and \(h=\exp(r^--r^+)\). Then we have
Corollary 3.
Relation (7) holds for any function \(r\in\mathscr{H}(1/2,\infty)\) with compact support.
The corollary immediately follows from the lemma, since both sides of equality (7) are continuous in the space \(\mathscr{H}(1/2,\infty)\).
Before passing to the proof of the lemma, let us make the following observation about the convergence of Fredholm determinants. Let \(K\) be a trace-class operator acting on a separable Hilbert space. Set \(|K|=\sqrt{K^*K}\). For \(l\ge 1\), let \(\bigwedge^{\!l}K\) stand for the exterior power of \(K\). The exterior powers are also of trace class and we have
Further, let \(K\) and \(K_n\) be trace-class operators, which may act on different Hilbert spaces. In order to establish the convergence
it is enough to verify the convergence
and the uniform estimate
Moreover, if \(K_n\) are positive self-adjoint operators, one can see from formula (8) that the condition
is sufficient for (9) to hold. Consider the discrete Hankel operator with a smooth symbol \(r\). Below we will use the simple estimate
For the continual Hankel operator with a smooth symbol \(r\), we will employ the similar estimate
Proof of the Lemma.
Recall that the functions \(R_n\) are given by formula (6). For their Sobolev seminorms we have the equality
whence the limit is
Similarly, for exterior powers of the operators \(\mathbf{H}(R_n)\), we have
The smoothness of function \(r\) immediately implies that \(\sup_{n\in\mathbb{N}}\|\widehat{R_n}\|_{L_1}<+\infty\). From the identity
we see that
which implies that
Assume now that we have two smooth functions \(r^{(1)}\), \(r^{(2)}\) with support lying in the interval \((-\pi,\pi)\). As above, we denote \(R_n^{(1)}(\theta)=r^{(1)}(n\theta)\), \(R_n^{(2)}(\theta)=r^{(2)}(n\theta)\). The proof of the limit relation
and the similar formula for the restricted Hankel operators
runs parallel to the considerations above. The Borodin–Okounkov–Geronimo–Case formula (5) concludes the proof of the lemma. \(\square\)
4. Scaling of Fredholm Determinants
It remains to express the Fredholm determinant on the left-hand side of formula (3) as the scaling limit of Toeplitz determinants.
Let \(f\) be a smooth function with compact support on \(\mathbb{R}\). As above, we let \(r(\,\cdot\,)=f(\,\cdot\,/(2\pi))\) and \(R_n(\theta)=r(n\theta)\).
Proposition 1.
We have
Let \(E\) be a complete separable metric space. Let \(\mu_n\) be a sequence of \(\sigma\)-finite Radon measures on \(E\) and \(K_n\) be a sequence of continuous kernels on \(E\). We assume that the kernels induce positive contractions on \(L_2(E, \mu_n)\), which, by a slight abuse of notation, we also denote by \(K_n\). Further, let \(\mu\) be a \(\sigma\)-finite Radon measure on \(E\) and let \(K\) be a continuous kernel, again inducing a positive contraction on \(L_2(E, \mu_n)\). We say that the sequence \(K_n\) \(F\)-converges to the limit kernel \(K\) if the following conditions hold:
-
1)
\(K_n\to K\) uniformly on compact subsets of \(E\times E\);
-
2)
measures \(K_n(x,x)\,d\mu_n(x)\) converge to \(K(x,x)\,d\mu(x)\) in total variation.
The definition immediately implies the following.
Proposition 2.
Assume that a sequence of continuous kernels \(K_n\) inducing nonnegative contractions \(F\)-converges to the limit kernel \(K\). Then for any compact set \(C\subset E\), we have that
Proof.
1. It is sufficient to verify the following equality for every \(l\):
and then use the uniform convergence of the series from the definition of Fredholm determinants.
2. Further, let \(\varepsilon>0\) and consider a compact subset \(C_\varepsilon=C\cap\{x\in E\colon K(x,x)\ge\varepsilon\}\). For \(x_1,\dots,x_n\in C_\varepsilon\), we have the uniform convergence
Note that the fractions in (10) are bounded from above by \(1\), since \(K_n\) are positive kernels. Using this fact, the convergence in total variation of the measures \(K_n(x,x)\,d\mu_n(x)\) to the measure \(K(x,x)\,d\mu(x)\), and then passing to the limit as \(\varepsilon\to 0\), we complete the proof. \(\square\)
Consider an interval \([a,b]\subset\mathbb{R}\). Proposition 1 immediately gives the following.
Corollary 4.
Let \(K\) be a trace-class operator on \(L_2([a, b])\) with a continuous kernel \(K(x, y)\). Also let \(K_n\) be a sequence of operators on \(\mathbb{Z}\) with a standard counting measure. Assume that for any \(x,y\in[a, b]\), the following limit relation holds:
where the convergence is uniform on \([a, b]\). Then we have
Now, Proposition 1 follows from formula (4) and formula (12) in Corollary 4.
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Acknowledgments
I am deeply grateful to Sergey Gorbunov and Roman Romanov for useful discussions and to the anonymous referee for the thorough review of the paper and the numerous improvements suggested.
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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2024, Vol. 58, No. 2, pp. 23–33 https://doi.org/10.4213/faa4203.
In loving memory of my teachers, Boris Markovich Gurevich (1938–2023) and Anatoly Moiseyevich Vershik (1933–2024)
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Bufetov, A. The Expectation of a Multiplicative Functional under the Sine-Process. Funct Anal Its Appl 58, 120–128 (2024). https://doi.org/10.1134/S0016266324020035
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DOI: https://doi.org/10.1134/S0016266324020035