Browsing by Subject "asymptotic expansion"

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    Dimer model: Full asymptotic expansion of the partition function
    (AIP, 2018) Bleher, Pavel; Elwood, Brad; Petrović, Dražen; Mathematical Sciences, School of Science
    We give a complete rigorous proof of the full asymptotic expansion of the partition function of the dimer model on a square lattice on a torus for general weights zh, zv of the dimer model and arbitrary dimensions of the lattice m, n. We assume m is even and we show that the asymptotic expansion depends on the parity of n. We review and extend the results of Ivashkevich et al. [J. Phys. A: Math. Gen. 35, 5543 (2002)] on the full asymptotic expansion of the partition function of the dimer model, and we give a rigorous estimate of the error term in the asymptotic expansion of the partition function.
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    On Random Polynomials Spanned by OPUC
    (2020-12) Aljubran, Hanan; Yattselev, Maxim; Bleher, Pavel; Mukhin, Evgeny; Roeder, Roland
    We consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.