Abstract
We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher-Hartwig singularities. This generalises two results: 1) a result of Berestycki, Webb and Wong [5] for root-type singularities, and 2) a result of Its and Krasovsky [37] for a Gaussian weight with a single jump-type singularity. We show that when we apply a piecewise constant thinning on the eigenvalues of a random Hermitian matrix drawn from a one-cut regular ensemble,
Results & Lemmas (7)
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Theorem 1.38
Theorem 1.38] (for more details, see the beginning of Section 4). In the present work, we restrict ourselves to the class of one-cut…
Theorem 1.38] (for more details, see the beginning of Section 4). In the present work, we restrict ourselves to the class of one-cut regular potentials whose equilibrium measure is supported on [−1, 1] instead of [a, b]. This is without loss of generality, as can be easily seen from a change of variables in (1.4), (1.11) and (1.12). For the reader’s convenience, we show that explicitly in Remark 1.4 below. An example of a one-cut regular potential is the Gaussian potential V (x) = 2x2. In this c
Theorem 1.1
Theorem 1.1 Let m ∈N, and let tj, αj and βj be such that tj ∈(−1, 1), tj ̸= tk for 1 ≤ j ̸= k ≤m, ℜαj > −1 and ℜβj ∈( −1 4, 1 4), for j =…
Theorem 1.1 Let m ∈N, and let tj, αj and βj be such that tj ∈(−1, 1), tj ̸= tk for 1 ≤ j ̸= k ≤m, ℜαj > −1 and ℜβj ∈( −1 4 , 1 4), for j = 1, ..., m. Let V be a one-cut regular potential whose equilibrium measure is supported on [−1, 1] with density ψ(x) √ 1 −x2, and let W : R →R be analytic in a neighbourhood of [−1, 1], locally H¨older-continuous on R and such that W(x) = O(V (x)), as |x| →∞. As n →∞, we have Dn(⃗α, ⃗β, V, W) = exp C1n2 + C2n + C3 log n + C4 + O log n
Theorem 1.1
Theorem 1.1 by integrating them. 3.1 Differential identity with respect to βk, k ∈ 1,..., m As mentioned in the outline, we will first…
Theorem 1.1 by integrating them. 3.1 Differential identity with respect to βk, k ∈{1, ..., m} As mentioned in the outline, we will first compute the asymptotics of Dn(⃗α, ⃗β, 2x2, 0), i.e. when the weight has the form w(x) = e−2nx2ω(x). For this we will need a differential identity with respect to βk, for each k ∈{1, ..., m}. Suppose f is a smooth and integrable function on R with sufficient decay at ±∞. The integral of f(x)ω(x) x−tk on R is not well-defined if ℜαk ≤0, even in the sense of principal v
Proposition 3.1
Proposition 3.1 (adapted from [42]). The regularised integral (3.6) satisfies Regk(f) = lim z→tk αk Z R f(x)ω(x) x −z dx −Jk(z) , (3.8)…
Proposition 3.1 (adapted from [42]). The regularised integral (3.6) satisfies Regk(f) = lim z→tk αk Z R f(x)ω(x) x −z dx −Jk(z) , (3.8) where the limit is taken along a non-tangential to the real line path in {z ∈C : ℑz > 0}, and Jk(z) =
Proposition 5.1
Proposition 5.1 As n →∞, log Dn(⃗α, ⃗β, 2x2, 0) Dn(⃗α,⃗0, 2x2, 0) = 2in m X j=1 arcsin tj + tj q 1 −t2 j βj + iA
Proposition 5.1 As n →∞, log Dn(⃗α, ⃗β, 2x2, 0) Dn(⃗α,⃗0, 2x2, 0) = 2in m X j=1 arcsin tj + tj q 1 −t2 j βj + iA
Proposition 6.1
Proposition 6.1 As n →∞, we have log Dn(⃗α, ⃗β, V, 0) Dn(⃗α, ⃗β, 2x2, 0) = −n2 2 Z 1 −1 p 1 −x2 V (x) −2x2 2 π + ψ(x) dx + n m X j=1
Proposition 6.1 As n →∞, we have log Dn(⃗α, ⃗β, V, 0) Dn(⃗α, ⃗β, 2x2, 0) = −n2 2 Z 1 −1 p 1 −x2 V (x) −2x2 2 π + ψ(x) dx + n m X j=1
Proposition 7.1
Proposition 7.1 As n →∞, log Dn(⃗α, ⃗β, V, W) Dn(⃗α, ⃗β, V, 0) = n Z 1 −1 ψ(x) p 1 −x2W(x)dx − 1 4π2 Z 1 −1 W(y) p
Proposition 7.1 As n →∞, log Dn(⃗α, ⃗β, V, W) Dn(⃗α, ⃗β, V, 0) = n Z 1 −1 ψ(x) p 1 −x2W(x)dx − 1 4π2 Z 1 −1 W(y) p
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