Mathematics

In this paper, we give a short Bayesian proof of Talagrand's celebrated majorizing-measure theorem (MMT). While the upper-bound direction of MMT follows relatively directly from standard arguments, the lower-bound direction is widely…

We study Bernoulli percolation on $\mathbb Z^d$ in dimensions ${d>6}$. We prove that a classical consequence of the van den Berg-Kesten inequality, often referred to as the Simon-Lieb inequality in the context of the Ising model, admits a…

The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $\mu$. We establish new sufficient conditions ensuring…

In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on $(n,k,L)$-systems. The main result is a proof of the Aljohani--Bamberg--Cameron conjecture which claims…

We derive closed form expressions for the lower expectations that correspond to total variation distance and chi-squared divergence balls around a probability mass function over a finite set.

We investigate integration by parts (IBP) formulae for stochastic Volterra equations and we establish the smoothing effect of the expectation. Due to the inherent path-dependent dynamics of this class of processes, standard…

Importance sampling (IS) consists in biasing samples from a distribution $f$ towards another distribution $g$. Concretely, given samples $X_i$ from $f$, the IS measure is $$\hat{g}_n = \frac{1}{Z_n}\sum_{i=1}^n \frac{g(X_i)}{f(X_i)}…

For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles…

Using the Baxter-Kelland-Wu coupling and the convergence of the height function of the six-vertex model to the Gaussian Free Field, we extract critical exponents for the planar critical random-cluster model at $q=4$, and the planar…

We use the machinery of a conditional probability space (R\'enyi, 1955) to obtain an Agreement Theorem (Aumann, 1976) under general conditions. A conditional probability space (CPS) is a family of probability measures defined relative to a…

We study the complex spectrum of the partial theta function \[ \Theta(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j, \qquad |q|<1, \] where a spectral value is a parameter for which \(\Theta(q,\cdot)\) has a multiple zero. Since the function is…

We consider $N\times N$ matrices $X$ with independent, identically distributed entries, and prove that the sequence of measures $\frac{ | \det (X-z)|^\gamma}{\mathbb{E}[ | \det (X-z)|^\gamma]}$ converge to the Gaussian Multiplicative Chaos…

We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this…

We consider Subtraction Nim, where two players have exactly the same options, but which is partizan in the sense that at the game ending, a partizan rule is applied for the decision of the winner. We consider the following example: Let $S$…

The spectrum of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which…

We study a question of Harju from 2019 regarding the existence of infinite ternary square-free words whose subsequences modulo $p$ and $q$ are also square-free for relatively prime integers $p$ and $q$. Among such pairs $(p, q)$ with $p, q…

We introduce variants of the Maker-Breaker and Waiter-Client games, which we call \emph{stotting}, in which a player grants a slight advantage to the opponent. We prove that a winning strategy in either stotting variant yields winning…

The $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials are studied in [Trans. Amer. Math. Soc. 364 (2012), 5491-5507], [Adv. Math. 229 (2012), 2123-2158], and [Stud. Appl. Math. 153 (2024), e12728], respectively, through…

In this paper, we prove a spectral restriction theorem on the three-dimensional Heisenberg nilmanifold. Since this manifold is an $\mathbb S^1$-bundle over the flat torus $\mathbb T^2$, the result provides a sub-elliptic counterpart of…

Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. The proof is based on a bespoke container…