Let $w_1, \dots, w_m$ be positive real weights whose sum is $1$, and let $v_1, \dots, v_m$ be i.i.d. Bernoulli$(p)$ random variables. If we let $X=\sum_{i=1}^m w_i v_i$, then we conjecture that for all $0\leq p\leq 1/3$ we have \[\mathbb{P}\big[X\geq \mathbb{E}[X]\big]\geq p.\] In this short note, we observe a connection of this conjecture with a version of the Manickam-Mikl\'os-Singhi conjecture, which allows one to prove it for sufficiently small values of $p$.
Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we consider a relaxed optimization problem over a set of sequences of ordered, but non-independent random variables. This allows us to reformulate it recursively as dynamic programming problem. The bound becomes an equality for the binary i.i.d.~random variables satisfying $\mathbb{P}\left[X_i=0\right]= 1-\frac{1}{t}$ and $\mathbb{P}\left[X_i=t\right]=\frac{1}{t}$, which remains the maximizer in the relaxed problem.
For each $s>0$, we consider a random Dirichlet series $X(s)=\sum_{k\geq 1}k^{-1/2-s}a_k\eta_k$, where $\eta_1$, $\eta_2,\ldots$ are independent and identically distributed random variables with mean zero and finite positive variance, and $(a_k)_{k\geq 1}$ is a deterministic sequence of real numbers satisfying $\sum_{k\geq 1}k^{-1-2s}a_k^2<\infty$ for each $s>0$ and $\sum_{k\geq 1}k^{-1}a_k^2=\infty$. We investigate the almost-sure fluctuations of $X(s)$ as $s\to0+$. Under these minimal assumptions, we construct examples exhibiting several non-standard forms of the law of the iterated logarithm (LIL) along suitable sequences: the normalization and the upper and lower limit constants may differ from their classical counterparts. We also show that a regular growth condition of the form $\sum_{k\leq n}k^{-1}a_k^2\sim c(\log n)^\beta$, where $c,\beta>0$, is not by itself sufficient to ensure a standard LIL. Finally, under an additional counting condition controlling the frequency of indices at which the weights $a_k$ are comparatively large, we prove that $(2{\rm Var}\,[X(s)]\log\log({\rm Var}\,[X(s)]))^{-1/2}X(s)$ has the almost-sure cluster set $[-1,1]$ as $s\to 0+$. The latter result is applied to several coefficient sequences of number-theoretic origin.
Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] Our proof tracks the total squared permanent of minors under successive row exposure. Up to $k=\lfloor n/2\rfloor$, the total squared permanent grows deterministically via the Boolean lattice up-operator; for larger $k$, the row exposure increments are governed by positive semidefinite Rademacher quadratic forms. Therefore, we confirms the exponential scale lower bound suggested by Tao and Vu.
Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1}\mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. We prove $\max_i X_i\leq_{\mathrm{st}}\max_i Z_i$, with equality in distribution if and only if $R=I_m$. We use this comparison to resolve the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex is the unique maximizer of the average probability of correct maximum-likelihood decoding at every signal-to-noise ratio. The same comparison proves the Simplex Mean Width Conjecture and gives the exact finite-energy performance of deterministic no-feedback AWGN codes with equiprobable messages, no restriction on the number of channel uses, and a maximal per-codeword energy constraint. The proof uses a Gaussian product inequality for log-concave functions whose first moments with respect to standard Gaussian measure vanish. A variational argument chooses one exponential tilt and one truncation endpoint in each coordinate so that this product inequality applies and a Gaussian change of measure returns all coordinates to the prescribed common threshold. A strict form of the product inequality also shows that, unless $R=I_m$, $\mathbb{P}\{X\leq c\mathbf{1}\}>\Phi(c)^m$ for every finite $c$, and hence gives the distributional equality statement. A Lean formalization is available at https://github.com/abhmul/weak-simplex-conjecture-lean.
Let $(X,Y)$ be independent standard real Gaussian random variables, and let $(P\in\mathbb C[X,Y])$ satisfy $\mathbb E!\left(P(X,Y)^m\right)=0$ for every $(m\ge1)$. Using the complex coordinates $Z=\frac{X+iY}{\sqrt2}, \qquad W=\frac{X-iY}{\sqrt2},$ we prove that the monomial support of $(P)$ is strictly one-sided with respect to the weight $\operatorname{wt}(Z^aW^b)=a-b$. Thus either every monomial occurring in $(P)$ has positive weight, or every monomial has negative weight. It follows that, for every $(Q\in\mathbb C[X,Y])$, $\mathbb E!\left(Q(X,Y)P(X,Y)^m\right)=0$ whenever $(m>\deg Q)$. This proves the two-variable Gaussian Moments Conjecture with the explicit threshold $m\ge\deg Q+1$. The main ingredient is a prime-isolation theorem for exposed faces of the Newton polygon of $(P)$. A $(p)$-adic valuation argument at moment indices of the form $(m=qp)$ separates the contribution of a chosen face from all remaining multinomial strata. Frobenius reduction then forces the constant terms of all positive powers of the associated one-variable Laurent polynomial to vanish. The theorem of Duistermaat and van der Kallen implies that the face has weights of one strict sign, while a planar convex-geometric argument rules out support containing weights of both signs. Combined with the known one-variable case and counterexamples in dimensions $(n\ge3)$, this determines the dimensions in which the Gaussian Moments Conjecture holds.
We study random Bernoulli measures $\mu_{\mathbf{p}}$ in a random environment $\mathbf{p}=(p_1,p_2,\ldots)$, where $p_n$ are independent and identically distributed on $(0,1)$. For almost every environment $\mathbf{p}$, we establish several almost sure properties of $\mu_{\mathbf{p}}$: its local dimension, $L^q$ dimensions, Rajchman property, mutual singularity with any fixed measure, and a dichotomy for normal numbers of its typical points.