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Persi Diaconis

Publications and source records attributed to Persi Diaconis.

At least 55 records · Page 3Linked to original sources
arXiv2017

The sample size required in importance sampling

The goal of importance sampling is to estimate the expected value of a given function with respect to a probability measure $ν$ using a random sample of size $n$ drawn from a different probability measure $μ$. If the two measures $μ$ and $ν$ are nearly singular with respect to each other, which is often the case in practice, the sample size required for accurate estimation is large. In this article it is shown that in a fairly general setting, a sample of size approximately $\exp(D(ν||μ))$ is necessary and sufficient for accurate estimation by importance sampling, where $D(ν||μ)$ is the Kullback-Leibler divergence of $μ$ from $ν$. In particular, the required sample size exhibits a kind of cut-off in the logarithmic scale. The theory is applied to obtain a general formula for the sample size required in importance sampling for one-parameter exponential families (Gibbs measures).

math.PR↗
arXiv2016

Probabilizing Parking Functions

We explore the link between combinatorics and probability generated by the question "What does a random parking function look like?" This gives rise to novel probabilistic interpretations of some elegant, known generating functions. It leads to new combinatorics: how many parking functions begin with $i$? We classify features (e.g., the full descent pattern) of parking functions that have exactly the same distribution among parking functions as among all functions. Finally, we develop the link between parking functions and Brownian excursion theory to give examples where the two ensembles differ.

math.PR↗
arXiv2016

A central limit theorem for a new statistic on permutations

This paper does three things: It proves a central limit theorem for novel permutation statistics (for example, the number of descents plus the number of descents in the inverse). It provides a clear illustration of a new approach to proving central limit theorems more generally. It gives us an opportunity to acknowledge the work of our teacher and friend B. V. Rao.

math.PR↗
arXiv2016

A. Hurwitz and the origins of random matrix theory in mathematics

The purpose of this article is to put forward the claim that Hurwitz's paper "Uber die Erzeugung der Invarianten durch Integration." [Gott. Nachrichten (1897), 71-90] should be regarded as the origin of random matrix theory in mathematics. Here Hurwitz introduced and developed the notion of an invariant measure for the matrix groups $SO(N)$ and $U(N)$. He also specified a calculus from which the explicit form of these measures could be computed in terms of an appropriate parametrisation - Hurwitz chose to use Euler angles. This enabled him to define and compute invariant group integrals over $SO(N)$ and $U(N)$. His main result can be interpreted probabilistically: the Euler angles of a uniformly distributed matrix are independent with beta distributions (and conversely). We use this interpretation to give some new probability results. How Hurwitz's ideas and methods show themselves in the subsequent work of Weyl, Dyson and others on foundational studies in random matrix theory is detailed.

math-ph↗
arXiv2015

Useful bounds on the extreme eigenvalues and vectors of matrices for Harper's operators

In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an $n\times n$ matrix of the form $M=C+D$ where $C$ is a circulant and $D$ a diagonal matrix. The discrete Schrödinger operators are an interesting special case. The Weyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of $C$ and $D$, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix $M$ tends to the harmonic oscillator on $L^2(\mathbb{R})$ and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending $M$ to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.

math.PR↗
arXiv2015

Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector $φ$. The goal of the paper is to give bounds on the amplitude $\max φ/\minφ$. Two approaches are proposed: one using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen in [7].

math.PR↗
arXiv2015

An Exercise (?) in Fourier Analysis on the Heisenberg Group

Let H(n) be the group of 3x3 uni-uppertriangular matrices with entries in Z/nZ, the integers mod n. We show that the simple random walk converges to the uniform distribution in order n^2 steps. The argument uses Fourier analysis and is surprisingly challenging. It introduces novel techniques for bounding the spectrum which are useful for a variety of walks on a variety of groups.

math.PR↗
arXiv2015

Central Limit Theorems for some Set Partition Statistics

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.

math.CO↗
arXiv2014

The mathematics of the flip and horseshoe shuffles

We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is "reversed", and then the cards are interlaced. Flip shuffles are when the reversal comes from flipping the half over so that we also need to account for face-up/face-down configurations while horseshoe shuffles are when the order of the cards are reversed but all cards still face the same direction. We show that these shuffles are closely related to faro shuffling and determine the order of the associated shuffling groups.

math.CO↗
arXiv2014

Universal Limit Theorems in Graph Coloring Problems With Connections to Extremal Combinatorics

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (non-normal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erd\H os and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.

math.PR↗
arXiv2014

On quantitative convergence to quasi-stationarity

The quantitative long time behavior of absorbing, finite, irreducible Markov processes is considered. Via Doob transforms, it is shown that only the knowledge of the ratio of the values of the underlying first Dirichlet eigenvector is necessary to come back to the well-investigated situation of the convergence to equilibrium of ergodic finite Markov processes. This leads to explicit estimates on the convergence to quasi-stationarity, in particular via functional inequalities. When the process is reversible, the optimal exponential rate consisting of the spectral gap between the two first Dirichlet eigenvalues is recovered. Several simple examples are provided to illustrate the bounds obtained.

math.PR↗
arXiv2014

Unseparated pairs and fixed points in random permutations

In a uniform random permutation Πof [n] := {1,2,...,n}, the set of elements k in [n-1] such that Π(k+1) = Π(k) + 1 has the same distribution as the set of fixed points of Πthat lie in [n-1]. We give three different proofs of this fact using, respectively, an enumeration relying on the inclusion-exclusion principle, the introduction of two different Markov chains to generate uniform random permutations, and the construction of a combinatorial bijection. We also obtain the distribution of the analogous set for circular permutations that consists of those k in [n] such that Π(k+1 mod n) = Π(k) + 1 mod n. This latter random set is just the set of fixed points of the commutator [ρ, Π], where ρis the n-cycle (1,2,...,n). We show for a general permutation ηthat, under weak conditions on the number of fixed points and 2-cycles of η, the total variation distance between the distribution of the number of fixed points of [η,Π] and a Poisson distribution with expected value 1 is small when n is large.

math.PR↗
arXiv2014

An introduction to multivariate Krawtchouk polynomials and their applications

Orthogonal polynomials for the multinomial distribution m(x, p) of N balls dropped into d boxes (box i has probability p(i)) are called multivariate Krawtchouk polynomials. This paper gives an introduction to their properties, collections of natural Markov chains which they explicitly diagonalize and associated bivariate multinomial distributions.

math.PR↗
arXiv2014

Fluctuations of the Bose-Einstein condensate

This article gives a rigorous analysis of the fluctuations of the Bose-Einstein condensate for a system of non-interacting bosons in an arbitrary potential, assuming that the system is governed by the canonical ensemble. As a result of the analysis, we are able to tell the order of fluctuations of the condensate fraction as well as its limiting distribution upon proper centering and scaling. This yields interesting results. For example, for a system of $n$ bosons in a 3D harmonic trap near the transition temperature, the order of fluctuations of the condensate fraction is $n^{-1/2}$ and the limiting distribution is normal, whereas for the 3D uniform Bose gas, the order of fluctuations is $n^{-1/3}$ and the limiting distribution is an explicit non-normal distribution. For a 2D harmonic trap, the order of fluctuations is $n^{-1/2}(\log n)^{1/2}$, which is larger than $n^{-1/2}$ but the limiting distribution is still normal. All of these results come as easy consequences of a general theorem.

math.PR↗
arXiv2013

Estimating and understanding exponential random graph models

We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017]. The theory explains a host of difficulties encountered by applied workers: many distinct models have essentially the same MLE, rendering the problems ``practically'' ill-posed. We give the first rigorous proofs of ``degeneracy'' observed in these models. Here, almost all graphs have essentially no edges or are essentially complete. We supplement recent work of Bhamidi, Bresler and Sly [2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2008) 803-812 IEEE] showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erdős-Rényi model. We also find classes of models where the limiting graphs differ from Erdős-Rényi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.

math.PR↗
arXiv2013

Some things we've learned (about Markov chain Monte Carlo)

This paper offers a personal review of some things we've learned about rates of convergence of Markov chains to their stationary distributions. The main topic is ways of speeding up diffusive behavior. It also points to open problems and how much more there is to do.

math.ST↗
arXiv2013

Combinatorics of balanced carries

We study the combinatorics of addition using balanced digits, deriving an analog of Holte's "amazing matrix" for carries in usual addition. The eigenvalues of this matrix for base b balanced addition of n numbers are found to be 1,1/b,...,1/b^n, and formulas are given for its left and right eigenvectors. It is shown that the left eigenvectors can be identified with hyperoctahedral Foulkes characters, and that the right eigenvectors can be identified with hyperoctahedral Eulerian idempotents. We also examine the carries that occur when a column of balanced digits is added, showing this process to be determinantal. The transfer matrix method and a serendipitous diagonalization are used to study this determinantal process.

math.CO↗