Motivated by a conjecture of Vaikuntanathan and Zamir, we study the pseudo-mixing of Kac's walk on $\mathrm{SO}(n)$: whether short trajectories are indistinguishable from Haar measure by low-complexity tests. We prove that the first $k$ columns mix in Wasserstein distance in $O(n(k+\log n)\log n)$ steps for fixed accuracy, resolving a conjecture of Oliveira. Combining this with a representation-theoretic variance bound, we show that if $T=\omega(nk(k+\log n)\log n)$, then every degree-$k$ polynomial normalized to have unit Haar variance has expectation under the $T$-step law within $o(1)$ of its Haar expectation. As an application, we show that this pseudo-mixing estimate can be used to prove the effectiveness of a fast Johnson--Lindenstrauss transform with the usual target dimension.
In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For odd $k$, the equilibrium measure is the uniform measure on $\mathfrak{A}_n$. For even $k$, the walk first mixes to the parity mixture determined by the hitting time, and in our range this mixture is asymptotically $U_{\mathfrak{S}_n}$. Our argument combines a refined fixed-time approximation for the random $k$-cycle walk near the cutoff window with an auxiliary marking scheme inspired by Jain-Sawhney's work (arXiv:2410.23944) on random transpositions. The main new feature is a parity-compatible coupling which handles both odd and even $k$-cycles in a unified framework. We also prove a hitting-time mixing result in the opposite regime $k\ge n-o(n^{1/2})$, and formulate a conjecture for all $2\le k\le n-1$.
Chen Shang, Jiahe Shen, Jiyue Zeng et al.· 1 citation
Let $X = (X_1, \ldots, X_n)$ be a random vector from any Borel probability law on $\mathbb{R}_+^n$. We revisit the problem of deriving a lower confidence bound (LCB) on a scalar parameter of that law. We recast classical work, beginning with Buehler, in purely probabilistic terms to form a more accessible and extensible framework. We then specialize the framework to the case where the components of $X$ are independent. In this context, we prove that Gaffke's bound is Buehler optimal for the order that it induces with respect to the maximum marginal mean parameter: $max_{i \in [n]} E_Q[X_i]$, which reduces to the common mean when the $X_i$ are independent and identically distributed. That is to say, no other valid LCB that orders samples in the same way as Gaffke's bound can improve on it with respect to this parameter.
Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This settles the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison for a normalized quadratic function of a hypergeometric random variable. The log-convexity inequality is strict for $p>1$ and $0<\theta<1$.
We give upper and lower bounds for the number of solutions of the equation $e_n(x,y) = g$ in the group $W_k=(C_p\wr C_{p^k})^2$, where $e_n(x,y)$ is the $n$-th Engel word and $g\in W_k$. We obtain several corollaries from this. First, we prove a stronger version of the Amit-Ashurst conjecture for Engel words in $W_k$. We also prove that Engel words are not probabilistic identities in profinite groups with arbitrarily large wreath product quotients $W_k$. To conclude, we construct closed subsets of $(C_p\wr\Z_p)^2$ with positive Haar measure, empty-interior, and which are the preimage of an Engel word map.
Let $k\ge1$ be an integer and let $\lambda$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $\lambda(n+h_1),\dots, \lambda(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average as well.
In the theory of mixing times, a famously wrong conjecture predicts that a sequence of Markov processes exhibits cutoff as soon as the product of their Poincar\'e constant and mixing time diverges. We prove that this statement becomes correct once the Poincar\'e constant $\gamma$ is replaced with its natural non-equilibrium refinement, which we denote by $\gamma_\star$. More precisely, we show that the width of the mixing window of any Markov process is $O(1/\gamma_\star)$. This estimate is sharp, and universal up to standard regularity assumptions: it holds on finite and infinite state spaces and from any initial condition, and it does not require reversibility, nor any kind of a chain rule. In addition, for deterministic initialization we show that $\gamma_\star\ge\kappa$, where $\kappa$ is the Bakry-\'Emery curvature, making our result broadly applicable. Finally, our proof is short and self-contained: we simply follow the classical idea of replacing the total variation distance by the more tractable $\chi^2$-divergence, but with the crucial novelty that the reference measure evolves in time, instead of being the equilibrium law.
Francesco Pedrotti, Justin Salez· 0 citations
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