On the asymptotics of a Toeplitz determinant with singularities

A. Its; I. Krasovsky; P. Deift

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On the asymptotics of a Toeplitz determinant with singularities

P. Deift, A. Its, I. Krasovsky

Functional Analysis · Submitted 6 Jun 2012

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The authors' main claim We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities.

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We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities. We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics. Our proof is based on the Riemann-Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants. The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher-Hartwig singularities and extensions to Hankel and Toeplitz+Hankel determinants in [15].

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities.	contribution 0.86	0.92	0.76	asserted 1.00
s1	We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics.	contribution 0.82	0.95	0.19	asserted 1.00
s2	Our proof is based on the Riemann-Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants.	method 1.00	0.84	0.06	asserted 1.00
s3	The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher-Hartwig singularities and extensions to Hankel and Toeplitz+Hankel determinants in [15].	interpretation 0.96	0.19	0.11	asserted 1.00

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Original abstract

We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities. We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics. Our proof is based on the Riemann-Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants. The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher-Hartwig singularities and extensions to Hankel and Toeplitz+Hankel determinants in [15].

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8 arXiv papers cite this in 29 sentences

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The first fact will be established through a variance estimate in the next section, where we make use of a Toeplitz determinant representation and results of [ref] .

The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase · The Circular Unitary Ensemble, the Main Result, and the Strategy of the Proof · 0.95
Following [ref] , where a similar situation arose, we proceed as follows.

Toeplitz determinants with merging singularities · RH problem for $R$ · 0.97
Third, the asymptotic formulas so obtained are substituted into the RHS of the differential identities, and the final result [ref] [ref] is obtained by integration.[ Note that if one just used the formula [ref] instead of the differential identities, then the product $∏_k=0^n_0χ_k^-2$ for small $n_0$, and hence the constant factor $E$ in [ref], would remain undetermined.] In [ref] , [ref] the authors also note various uniformity properties of the large $n$ asymptotics for Toeplitz determinants in the parameters, such as $_j$, $β_j$, on which $f$ depends.

Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results · Fisher-Hartwig singularities. Fisher-Hartwig conjecture · 0.96

Cites it as support 1 paper

One can show that the error term is uniform in compact subsets of ${z_i≠z_j}$: see e.g. [ref] .

The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase · Asymptotics of $D_n-1(σ_2)$ · 0.99

Cites it as background 4 papers

We then review recent results from [ref] on asymptotics of such Toeplitz determinants.

The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase · Introduction · 0.92
Asymptotics for Toeplitz determinants with Fisher-Hartwig singularities have been studied by many authors [ref] .

Asymptotics for Toeplitz determinants: perturbation of symbols with a gap · Introduction · 0.99
The large $n$ behavior of Toeplitz determinants with several FH singularities has been studied by many authors under various assumptions on $V$ and the values of the parameters $α_j$ and $β_j$, see, e.g., [ref] .

Toeplitz determinants with merging singularities · Introduction · 1.00

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