arXiv subjects

Ivan Z. Feng

Publications and source records attributed to Ivan Z. Feng.

6 recordsLinked to original sources

arXiv2026

The dual Burnside process

The Burnside process is a classical Markov chain for sampling uniformly from group orbits. We give a systematic study of the dual Burnside process, obtained by interchanging the roles of group elements and states. This dual chain has stationary law $π(g)\propto |X_g|$, is reversible, and admits a matrix factorization $Q=AB$, $K=BA$ with the classical Burnside kernel $K$. As a consequence, the two chains share all nonzero eigenvalues and have mixing times that differ by at most one step. We further establish universal Doeblin floors, orbit- and conjugacy-class lumpings, exact stabilizer/fixed-set quotient pairs, and transfer principles between $Q$ and $K$. We analyze the explicit examples of the value-permutation model $S_k$ acting on $[k]^n$ and the coordinate-permutation model $S_n$ acting on $[k]^n$. In the value-permutation model, for fixed $k\ge3$, the dual fixed-symbol-set quotient has $2^k-k-1$ states, independent of $n$, preserves the full nonzero spectrum, and has limiting nontrivial spectral radius $1/2$. These results show that the dual chain provides both a conceptual mirror to the classical Burnside process and a genuinely useful compression mechanism for symmetry-aware Markov chain Monte Carlo.

math.PR↗

arXiv2026

Analysis of a twisted Bose-Einstein Markov chain with applications to sampling Catalan structures

We analyze a twisted Bose-Einstein Markov chain on $[k]^n$ arising from the Diaconis-Zhong twisted Burnside process. We derive explicit formulas for the transition kernel and stationary distribution of both the original chain and its lumped process, and we identify this chain as a special case of Crane's cut-and-paste process. We establish bounds on the mixing time and determine the second-largest eigenvalue of the original chain and its lumped process. Complementing the fixed-$k$ cutoff theorem of Crane and Lalley, our bounds show that both chains mix in $Θ(\log n)$ steps when $k$ grows as a fixed positive power of $n$. For the non-twisted Bose-Einstein Markov chain, this resolves a conjecture of Diaconis. As an application, we study the Burnside processes on parking functions and labeled Dyck paths, which give novel Markov chain Monte Carlo algorithms for sampling an increasing parking function and a Dyck path, respectively, approximately uniformly at random. We show that these chains are rapidly mixing, with mixing times of $Θ(\log n)$.

math.PR↗

arXiv2026

Cutoff profiles for colored top-m-to-random shuffles with growing block size

We study the $p$-colored top-$m$-to-random shuffle on $C_p\wr S_n$ when the block size $m=m_n$ grows with $n$. Let $E_{k_n}^{(m_n)}$ be the number of labels never touched after $k_n$ independent uniform $m_n$-subset draws, and set $b_n=n-m_n$, $q_n=b_n/n$, and $λ_n=nq_n^{k_n}$. We prove that if $λ_n\toλ\in(0,\infty)$ and $b_n\to\infty$, then $E_{k_n}^{(m_n)}\Rightarrow\mathrm{Poisson}(λ)$. Combining this with the exact nested-set reduction for colored top-$m$-to-random shuffles, we obtain growing-block total variation, separation, and integrated likelihood-ratio profiles. In particular, if $Q_{n,p}^{(m_n)}$ is the one-step law and $U_{n,p}$ is uniform on $C_p\wr S_n$, then the separation distance from $(Q_{n,p}^{(m_n)})^{*k_n}$ to $U_{n,p}$ tends to $1-e^{-λ}(1+λ)$ for $p=1$ and to $1-e^{-λ}$ for $p\ge2$. The criterion applies to small blocks, proportional blocks, and near-full blocks.

math.PR↗

arXiv2026

Explicit cutoff profiles for colored top-$m$-to-random shuffles

We study $p$-colored top-$m$-to-random on the wreath product $G_{n,p}=C_p\wr S_n$, with $m$ fixed. Using the Nakano-Sadahiro-Sakurai basis elements $B_m$, we obtain exact nested-set occupancy mixtures and reduce the likelihood ratio to the single statistic $L_p$. This yields exact formulas for separation and $L^\infty(U)$, and exact one-dimensional formulas for total variation, $L^q(U)$ ($1\le q<\infty$), $χ^2$, and relative entropy. At $k=\Bigl\lfloor \frac{n}{m}(\log n+c)\Bigr\rfloor$, the number of never-chosen labels in the associated $m$-subset occupancy model converges in law to $\mathrm{Poisson}(e^{-c})$, giving the total-variation profile $f_p(c)$, the separation profile, and the corresponding $L^q(U)$, $L^\infty(U)$, $χ^2$, and relative-entropy profiles. For $m=1$ we recover colored top-to-random; for $p=1$, the total-variation profile reduces to the Diaconis-Fill-Pitman profile. For the reversed chain, we also identify optimal strong stationary times whose tail probabilities are exactly the separation distances.

math.PR↗

arXiv2026

The Math Teaching Atlas: Trails, Anchors, and Compass in Action

Mathematics is a mountain, but students need more than descriptions of the view: they need a trail they can actually walk. This paper presents the Math Teaching Atlas, a framework for mathematical exposition built around route units (single steps with explicit justifications), routeways (chains of route units), and roadmaps of multiple valid routeways to the same destination. New ideas are introduced through familiar anchors, while a mathematical compass (motivation) and driving simulations (concrete examples) help students extend known routeways and construct new ones. The paper also develops a route geometry on pedagogical route graphs, thereby beginning a mathematization of mathematics pedagogy itself. Taken together, the framework aims to make mathematical explanations more visible, navigable, and teachable.

math.HO↗

arXiv2025

Periodicity Uncovered: A Deep Dive into Bott's Theorems in K-Theory and Fiber Bundles

This paper presents a comparison between two versions of Bott Periodicity Theorems: one in topological K-theory and the other in stable homotopy groups of classical groups. It begins with an introduction to K-theory, discussing vector bundles and their role in understanding the algebraic and topological aspects of these spaces. Then the two versions of Bott periodicity, as well as the topological notions necessary to understand them, are further explored. The aim is to illustrate the connections and distinctions between these two theorems, deepening our understanding of their underlying mathematical structures such as topological K-theory and fiber bundles.

math.AT↗