Related papers: Local tail bounds for polynomials on the discrete …

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be…

A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…

We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least $l^2$, for large $l$. The…

This paper is the Part II of a serious work about T product tensors focusing at establishing new probability bounds for sums of random, independent, T product tensors. These probability bounds characterize large deviation behavior of the…

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two…

We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{H}$ in $\mathbb{G}(n,p)$, the sparse Erd\H{o}s-R\'enyi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{-1/\Delta}…

The paper studies the probability for a Galois group of a random polynomial to be $A_n$. We focus on the so-called large box model, where we choose the coefficients of the polynomial independently and uniformly from $\{-L,\ldots, L\}$. The…

We study the local limit theorem for weighted sums of Bernoulli variables. We show on examples that this is an important question in the general theory of the local limit theorem, and which turns up to be not well explored. The examples we…

We consider the massive scalar field equation $\Box_{g_{RN}} \phi = m^2 \phi$ on any subextremal Reissner--Nordstr\"{o}m exterior metric $g_{RN}$. We prove that solutions with localized initial data decay pointwise-in-time at the polynomial…

We consider pairs of toral automorphisms (A,B) satisfying an invariant cone property. At each iteration, A acts with probability p and B with probability 1-p. We prove exponential decay of correlations for a class of Holder continuous…

In this paper I prove a conjecture which gives a lower bound for the largest absolute value of the coefficients of the n-th cyclotomic polynomial for some n. Moreover this estimate is essentially sharp.

We obtain first decay rates of probabilities of tails of multivariate polynomials built on independent random variables with heavy tails. Then we derive stable limit theorems for nonconventional sums of the form $\sum_{Nt\geq n\geq…

In high dimensional percolation at parameter $p < p_c$, the one-arm probability $\pi_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $\pi_p(n) / \pi_{p_c}(n)$, establishing a form…

We consider Glauber dynamics reversible with respect to Gibbs measures with heavy tails. Spins are unbounded. The interactions are bounded and finite range. The self potential enters into two classes of measures, $\kappa$-concave…

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic…

A short note on bounds on distance to variety of a point in terms of the Taylor coefficients at the point.

We obtain an asymptotic expansion for the tails of the random variable $\tcal=\arg\max_{u\in\mathbb{R}}(\mathcal{A}_2(u)-u^2)$ where $\mathcal{A}_2$ is the Airy$_2$ process. Using the formula of Schehr \cite{Sch} that connects the density…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

Assume that the Riemann hypothesis holds for Dedekind zeta functions. Under this assumption, we prove that a degree $d$ polynomial with random multiplicative $\pm1$ coefficients is irreducible in $\mathbb{Z}[x]$ with probability…

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that $n$-variate polynomials of total degree at most $d$ over grids, i.e. sets of the form $A_1 \times A_2 \times \cdots \times A_n$, form error-correcting codes (of distance at least…