Related papers: On the limiting behaviour of arithmetic toral eige…

We study asymptotics of the dimer model on large toric graphs. Let $\mathbb L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$ be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we show…

The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…

We consider the random normal matrices with quadratic external potentials where the associated orthogonal polynomials are Hermite polynomials and the limiting support (called droplet) of the eigenvalues is an ellipse. We calculate the…

Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$ such as…

The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero,…

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times…

We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…

Let f be a sum of exponentials of the form exp(2 pi i N x), where the N are distinct integers. We call f an idempotent trigonometric polynomial (because the convolution of f with itself is f) or, simply, an idempotent. We show that for…

We consider random systems of equations over the reals, with $m$ equations and $m$ unknowns $P_i(t)+X_i(t)=0$, $t\in\mathbb{R}^m$, $i=1,...,m$, where the $P_i$'s are non-random polynomials having degrees $d_i$'s (the "signal") and the…

In this paper, we construct families of polynomials defined by recurrence relations related to mean-zero random walks. We show these families of polynomials can be used to approximate $z^n$ by a polynomial of degree $\sim \sqrt{n}$ in…

In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues $\lambda_1(B_r)$ of ``balls'' $B_r:=\{\rho<r\}$ on a complete Riemannian manfold $M$ as $r\rightarrow +\infty$, where…

This paper investigates the asymptotic behavior of the principal eigenvalue $\lambda(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -\Delta_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c…

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f=y^n+\sum_{i=0}^{n-1}a_i(x)y^i\in\mathbb F_q[x][y]$ with i.i.d coefficients $a_i$ taking values in the set…

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related…

Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under…

We introduce several notions of `random fewnomials', i.e. random polynomials with a fixed number f of monomials of degree N. The f exponents are chosen at random and then the coefficients are chosen to be Gaussian random, mainly from the…

The Painleve-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers m and n. These functions have applications to nonlinear wave equations,…

For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$ of degree $k$, a family of polynomials $F_m(x)$ depending on $f(x)$ is constructed with the properties: (i) there is exactly one irreducible factor $\Phi_{d,f}(x)$…

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. In this paper, we compute the Sato-Tate group of $J_m$. Currently,…

The trigonometric monomial $\cos(\left\langle k, x \right\rangle)$ on $\mathbb{T}^d$, a harmonic polynomial $p: \mathbb{S}^{d-1} \rightarrow \mathbb{R}$ of degree $k$ and a Laplacian eigenfunction $-\Delta f = k^2 f$ have root in each ball…