About

Persi Diaconis is the Mary V. Sunseri Professor in the School of Humanities and Sciences and a Professor of Mathematics at Stanford University. He is a mathematician and statistician working in probability, combinatorics, and group theory with a focus on applications to statistics and scientific computing. A specialty of his is the rates of convergence of Markov chains. His current interests include adapting many mathematical developments to provide useful insights to practitioners in large real-world simulations.

Research topics

• Computer Science
• Mathematics
• Sociology
• Mathematical economics
• Artificial Intelligence
• Art
• Epistemology
• Philosophy
• History
• Classics
• Literature
• Mathematical optimization
• Algorithm
• Library science
• Arithmetic

Selected publications

• A curiously slowly mixing Markov chain

  ArXiv.org · 2025-11-03

  preprintOpen access1st authorCorresponding

  We study a Markov chain with very different mixing rates depending on how mixing is measured. The chain is the "Burnside process on the hypercube $C_2^n$." Started at the all-zeros state, it mixes in a bounded number of steps, no matter how large $n$ is, in $\ell^1$ and in $\ell^2$. And started at general $x$, it mixes in at most $\log n$ steps in $\ell^1$. But, in $\ell^2$, it takes $\frac{n}{\log n}$ steps for most starting $x$. The $\ell^2$ mixing results follow from an explicit diagonalization of the Markov chain into binomial-coefficient-valued eigenvectors.

  PublisherOA PDFDOI

• Probabilizing semigroups???

  Semigroup Forum · 2025-08-04

  article1st authorCorresponding
  PublisherDOI

• Counting the number of group orbits by marrying the Burnside process with importance sampling

  ArXiv.org · 2025-01-20 · 1 citations

  preprintOpen access1st authorCorresponding

  This paper introduces a novel and general algorithm for approximately counting the number of orbits under group actions. The method is based on combining the Burnside process and importance sampling. Specializing to unitriangular groups yields an efficient algorithm for estimating the number of conjugacy classes of such groups.

  PublisherOA PDFDOI

• Random sampling of contingency tables and partitions: Two practical examples of the Burnside process

  Statistics and Computing · 2025-08-30 · 2 citations

  articleOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Poisson approximation for large permutation groups

  Advances in Applied Mathematics · 2025-03-30 · 3 citations

  article1st author
  PublisherDOI

• Markov chains on Weyl groups from the geometry of the flag variety

  ArXiv.org · 2025-10-02

  preprintOpen access1st authorCorresponding

  This paper studies a basic Markov chain, the Burnside process, on the space of flags $G/B$ with $G = GL_n(\mathbb{F}_q)$ and $B$ its upper triangular matrices. This gives rise to a shuffling: a Markov chain on the symmetric group realized via the Bruhat decomposition. Actually running and describing this Markov chain requires understanding Springer fibers and the Steinberg variety. The main results give a practical algorithm for all n and q and determine the limiting behavior of the chain when q is large. In describing this behavior, we find interesting connections to the combinatorics of the Robinson-Schensted correspondence and to the geometry of orbital varieties. The construction and description is then carried over to finite Chevalley groups of arbitrary type, describing a new class of Markov chains on Weyl groups.

  PublisherOA PDFDOI

• Counting the number of group orbits by marrying the Burnside process with importance sampling

  Advances in Applied Mathematics · 2025-08-22 · 3 citations

  article1st author
  PublisherDOI

• Permuton and local limits for the Luce model

  ArXiv.org · 2025-09-09

  preprintOpen accessSenior author

  We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the weights, we establish a permuton limit theorem describing the global behavior of Luce-distributed permutations and derive an explicit density of the limiting permuton. We further compute limiting pattern densities and analyze the differences between exact Luce permutations and their permuton approximations. We also study the local convergence of these permutations, proving a quenched Benjamini--Schramm limit and a central limit theorem for consecutive pattern occurrences. Finally, we prove a central limit theorem for the number of inversions.

  PublisherOA PDFDOI

• Estimating the size of a set using cascading exclusion

  ArXiv.org · 2025-08-07

  preprintOpen access

  Let $S$ be a finite set, and $X_1,\ldots,X_n$ an i.i.d. uniform sample from $S$. To estimate the size $|S|$, without further structure, one can wait for repeats and use the birthday problem. This requires a sample size of the order $|S|^\frac{1}{2}$. On the other hand, if $S=\{1,2,\ldots,|S|\}$, the maximum of the sample blown up by $n/(n-1)$ gives an efficient estimator based on any growing sample size. This paper gives refinements that interpolate between these extremes. A general non-asymptotic theory is developed. This includes estimating the volume of a compact convex set, the unseen species problem, and a host of testing problems that follow from the question `Is this new observation a typical pick from a large prespecified population?' We also treat regression style predictors. A general theorem gives non-parametric finite $n$ error bounds in all cases.

  PublisherOA PDFDOI

• A Vershik-Kerov theorem for wreath products

  arXiv (Cornell University) · 2024-08-08

  preprintOpen accessSenior author

  Let $G_{n,k}$ be the group of permutations of $\{1,2,\ldots, kn\}$ that permutes the first $k$ symbols arbitrarily, then the next $k$ symbols and so on through the last $k$ symbols. Finally the $n$ blocks of size $k$ are permuted in an arbitrary way. For $σ$ chosen uniformly in $G_{n,k}$, let $L_{n,k}$ be the length of the longest increasing subsequence in $σ$. For $k,n$ growing, we determine that the limiting mean of $L_{n,k}$ is asymptotic to $4\sqrt{nk}$. This is different from parallel variations of the Vershik-Kerov theorem for colored permutations.

  PublisherOA PDFDOI

Recent grants

• The Mathematics of Mixing

  NSF · $450k · 2012–2016

• New Techniques and Analyses for Random Sampling

  NSF · $575k · 2020–2026

Frequent coauthors

• Laurent Saloff‐Coste

  Cornell University

  59 shared

• Susan Holmes

  Stanford University

  48 shared

• Sourav Chatterjee

  40 shared

• Laurent Miclo

  Institut de Mathématiques de Toulouse

  32 shared

• Kshitij Khare

  30 shared

• Ron Graham

  University of California, San Diego

  28 shared

• David A. Freedman

  24 shared

• Jason Fulman

  19 shared

Labs

• Persi Diaconis's LabPI

  Not provided

Education

• B.A., Mathematics

  Harvard University

  1969

• Ph.D., Statistics

  Stanford University

  1974

Awards & honors

• Stanford Honors Thesis Prizes - Symbolic Systems
• Glushko Prize for Excellence in Undergraduate Research in Sy…
• Barwise Award for Distinguished Contributions to Symbolic Sy…
• Symbolic Systems Distinguished Teaching Award