Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results

Theorem 1. Let ϕ eiθ
> 0 be a continuous, positive function on the unit circle S1 and let Dn(ϕ) be the associated Toeplitz determinant. Then (6) lim n→∞ 1 n log Dn(ϕ) = 1 2π Z π −π log ϕ  eiθ

Theorem 2. Let ϕ eiθ
be a real-valued function in L∞(S1) with m ≤ϕ eiθ
≤M a.e. If F(λ) is any continuous function defined on the interval m ≤λ ≤M, we have (8) lim n→∞ F  λ(n) 1  + · · · + F

Theorem 3 (Szeg˝o Strong Limit Theorem). Let ϕ eiθ
be a positive, C1+ε, ε > 0, function on S1. Let (log ϕ)k = 1 2π R π −π e−ikθ log ϕ eiθ
dθ, k ∈Z, denote the Fourier coefficients of log ϕ eiθ
. Then (35) lim n→∞

Theorem 4 (Ibragimov). Let ϕ eiθ
dθ 2π be a probability measure on S1 and suppose that log ϕ eiθ
is integrable. Then (35) is always true in the following sense: limn→∞ Dn(ϕ) en(log ϕ)0 always exists and equals eE(ϕ), including the case where one, and hence both, are infinite. □ One can consider Toeplitz matrices Tn(dµ) for general probability measures dµ(θ) = ϕ eiθ
dθ 2π + dµs(θ) on S1, where dµs denotes the singular part of dµ. We have Tn(dµ) = {µj−k}0≤j, k≤n−1, where µℓ= R

Theorem 1.

Theorem 5 ([Sz1][Sz2][Sz4][Ver]). Let dµ(θ) = ϕ eiθ
dθ 2π + dµs(θ) be a probability measure as above. Then Szeg˝o’s Limit Theorem is always true in the following sense: limn→∞Dn(dµ) 1 n always exists and equals exp hR π −π log ϕ eiθ
dθ 2π i , including the case where one, and hence both, are zero. □ Szeg˝o proved the result when dµs = 0, whereas Verblunsky was able to handle the case dµs ̸= 0.

Theorem 6 ([GolIbr]). Let dµ(θ) = ϕ eiθ
dθ 2π + dµs(θ) be a probability measure as above, and suppose log ϕ eiθ
∈L1 dθ 2π
. If dµs ̸= 0, then (38) lim n→∞ Dn(dµ) exp h

Theorem 7. Let V (eiθ) ∈L1(S1) be a (possibly complex-valued) function on S1 with Fourier coefficients (Vk)k∈Z satisfying (86) ∞ X k=−∞ |k| |Vk|2 < ∞, then (87) lim n→∞ Dn eV
enV0 = e

Corollary 10.42 of [BottSilb3]. 22

Theorem 9. [DIK1] Conjecture 8 holds. The relation of the semi-norm |||β||| to the minimization problem is given by the following elementary result.

Lemma 10. [DIK1] Given β, there are only two, mutually exclusive, possibilities: Either (i) There exists bβ ∈Oβ such that |||bβ||| < 1. Then such a bβ is unique and it is the unique element of Mβ = n bβ o or (ii) There exists bβ ∈Oβ such that |||bβ||| = 1. Then there are at least two such bβ’s and all of them are obtained from each other by a repeated application of the following rule: add 1 to a bβj with the smallest real part, ℜbβj = mink ℜbβk, and subtract 1 from a bβj′ with the largest real

Theorem 9 can also be used to analyze the asymptotic behavior of the eigenvalues λ(n) 1 ≤λ(n) 2 ≤ · · · ≤λ(n) n , n →∞, of a Toeplitz matrix Tn(ϕ) with a real symbol ϕ. At the beginning of our story (see Theorem 2) we interpreted Szeg˝o’s early results as saying that the λ(n) j ’s were equidistributed as n →∞. But what about the behavior of individual eigenvalues, λ(n) k , n →∞? Early work on this question considered the so-called extreme eigenvalues λ(n) k and λ(n)