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## Short cycles in random permutations

Previously we showed that the distribution of fixed points of a random permutation of $n$ elements behaves asymptotically (in the limit as $n \to \infty$) like a Poisson random variable with parameter $\lambda = 1$. As it turns out, this generalizes to the following.

Theorem: As $n \to \infty$, the number of cycles of length $1, 2, ... k$ of a random permutation of $n$ elements are asymptotically independent Poisson with parameters $1, \frac{1}{2}, ... \frac{1}{k}$.

This is a fairly strong statement which essentially settles the asymptotic description of short cycles in random permutations.

## Fixed points of random permutations

The following two results are straightforward and reasonably well-known exercises in combinatorics:

1. The number of permutations on $n$ elements with no fixed points (derangements) is approximately $\frac{n!}{e}$.
2. The expected number of fixed points of a random permutation on $n$ elements is $1$.

As it turns out, it is possible to say substantially more about the distribution of fixed points of a random permutation. In fact, the following is true.

Theorem: As $n \to \infty$, the distribution of the number of fixed points of a random permutation on $n$ elements is asymptotically Poisson with rate $\lambda = 1$.

## Moments, Hankel determinants, orthogonal polynomials, Motzkin paths, and continued fractions

Previously we described all finite-dimensional random algebras with faithful states. In this post we will describe states on the infinite-dimensional $^{\dagger}$-algebra $\mathbb{C}[x]$. Along the way we will run into and connect some beautiful and classical mathematical objects.

A special case of part of the following discussion can be found in an old post on the Catalan numbers.

## Lower bounds on off-diagonal Ramsey numbers

The goal of this post is to prove the following elementary lower bound for off-diagonal Ramsey numbers $R(s, t)$ (where $s \ge 3$ is fixed and we are interested in the asymptotic behavior as $t$ gets large):

$\displaystyle R(s, t) = \Omega \left( \left( \frac{t}{\log t} \right)^{ \frac{s}{2} } \right).$

The proof does not make use of the Lovász local lemma, which improves the bound by a factor of $\left( \frac{t}{\log t} \right)^{ \frac{1}{2} }$; nevertheless, I think it’s a nice exercise in asymptotics and the probabilistic method. (Also, it’s never explicitly given in Alon and Spencer.)

## The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation by hand, but it is conceptually cleaner to use the symmetries of the situation and representation theory. Deducing these results using representation theory has the added benefit that it identifies which parts of the situation depend only on symmetry and which parts depend on the particular form of the Hamiltonian. This is nicely explained in Singer’s Linearity, symmetry, and prediction in the hydrogen atom.

For awhile now I’ve been interested in finding a toy model to study the basic structure of the arguments involved, as well as more generally to get a hang for quantum mechanics, while avoiding some of the mathematical difficulties. Today I’d like to describe one such model involving finite graphs, which replaces the infinite-dimensional Hilbert spaces and Lie groups occurring in the analysis of the hydrogen atom with finite-dimensional Hilbert spaces and finite groups. This model will, among other things, allow us to think of representations of finite groups as particles moving around on graphs.

## Update, and the combinatorics of quintic equations

A brief update. I’ve been at Cambridge for the last week or so now, and lectures have finally started. I am, tentatively, taking the following Part II classes:

• Riemann Surfaces
• Topics in Analysis Probability and Measure
• Graph Theory
• Linear Analysis (Functional Analysis)
• Logic and Set Theory

I will also attempt to sit in on Part III Algebraic Number Theory, and I will also be self-studying Part II Number Theory and Galois Theory for the Tripos.

As far as this blog goes, my current plan is to blog about interesting topics which come up in my lectures and self-study, partly as a study tool and partly because there are a few big theorems I’d like to get around to understanding this year and some of the material in my lectures will be useful for those theorems.

Today I’d like to blog about something completely different. Here is a fun trick the first half of which I learned somewhere on MO. Recall that the Abel-Ruffini theorem states that the roots of a general quintic polynomial cannot in general be written down in terms of radicals. However, it is known that it is possible to solve general quintics if in addition to radicals one allows Bring radicals. To state this result in a form which will be particularly convenient for the following post, this is equivalent to being able to solve a quintic of the form

$\displaystyle y = 1 + xy^5$

for $y$ in terms of $x$. It just so happens that a particular branch of the above function has a particularly nice Taylor series; in fact, the branch analytic in a neighborhood of the origin is given by

$\displaystyle y = \sum_{n \ge 0} \frac{1}{4n+1} {5n \choose n} x^n$.

This should remind you of the well-known fact that the generating function $\displaystyle y = \sum_{n \ge 0} \frac{1}{n+1} {2n \choose n} x^n$ for the Catalan numbers satisfies $y = 1 + xy^2$. In fact, there is a really nice combinatorial proof of the following general fact: the generating function $\displaystyle y = \sum_{n \ge 0} \frac{1}{(k-1)n+1} {kn \choose n} x^n$ satisfies

$y = 1 + xy^k$.

## Test your intuition: consecutive tails

Something very unfortunate has happened: several things I have recently written that could have been blog entries are instead answers on math.SE! In the interest of exposition beyond the Q&A format I am going to “rescue” one of these answers. It is an answer to the following question, which I would like you to test your intuition about:

Flip $150$ coins. What is the probability that, at some point, you flipped at least $7$ consecutive tails?

Jot down a quick estimate; see if you can get within a factor of $2$ or so of the actual answer, which is below the fold.

## Walks on graphs and statistical mechanics

I finally learned the solution to a little puzzle that’s been bothering me for awhile.

The setup of the puzzle is as follows. Let $G$ be a weighted undirected graph, e.g. to each edge $i \leftrightarrow j$ is associated a non-negative real number $a_{ij}$, and let $A$ be the corresponding weighted adjacency matrix. If $A$ is stochastic, one can interpret the weights $a_{ij}$ as transition probabilities between the vertices which describe a Markov chain. (The undirected condition then means that the transition probability between two states doesn’t depend on the order in which the transition occurs.) So one can talk about random walks on such a graph, and between any two vertices the most likely walk is the one which maximizes the product of the weights of the corresponding edges.

Suppose you don’t want to maximize a product associated to the edges, but a sum. For example, if the vertices of $G$ are locations to which you want to travel, then maybe you want the most likely random walk to also be the shortest one. If $E_{ij}$ is the distance between vertex $i$ and vertex $j$, then a natural way to do this is to set

$a_{ij} = e^{- \beta E_{ij}}$

where $\beta$ is some positive constant. Then the weight of a path is a monotonically decreasing function of its total length, and (fudging the stochastic constraint a bit) the most likely path between two vertices, at least if $\beta$ is sufficiently large, is going to be the shortest one. In fact, the larger $\beta$ is, the more likely you are to always be on the shortest path, since the contribution from any longer paths becomes vanishingly small. As $\beta \to \infty$, the ring in which the entries of the adjacency matrix lives stops being $\mathbb{R}$ and becomes (a version of) the tropical semiring.

That’s pretty cool, but it’s not what’s been puzzling me. What’s been puzzling me is that matrix entries in powers of $A$ look an awful lot like partition functions in statistical mechanics, with $\beta$ playing the role of the inverse temperature and $E_{ij}$ playing the role of energies. So, for awhile now, I’ve been wondering whether they actually are partition functions of systems I can construct starting from the matrix $A$. It turns out that the answer is yes: the corresponding systems are called one-dimensional vertex models, and in the literature the connection to matrix entries is called the transfer matrix method. I learned this from an expository article by Vaughan Jones, “In and around the origin of quantum groups,” and today I’d like to briefly explain how it works.

## Hecke algebras and the Kazhdan-Lusztig polynomials

The Hecke algebra attached to a Coxeter system $(W, S)$ is a deformation of the group algebra of $W$ defined as follows. Take the free $\mathbb{Z}[q^{ \frac{1}{2} }, q^{ - \frac{1}{2} }]$-module $\mathcal{H}_W$ with basis $T_w, w \in W$, and impose the multiplicative relations

$T_w T_s = T_{ws}$

if $\ell(sw) > \ell(w)$, and

$T_w T_s = q T_{ws} + (q - 1) T_w$

otherwise. (For now, ignore the square root of $q$.) Humphreys proves that these relations describe a unique associative algebra structure on $\mathcal{H}_W$ with $T_e$ as the identity, but the proof is somewhat unenlightening, so I will skip it. (Actually, the only purpose of this post is to motivate the definition of the Kazhdan-Lusztig polynomials, so I’ll be referencing the proofs in Humphreys rather than giving them.)

The motivation behind this definition is a somewhat long story. When $W$ is the Weyl group of an algebraic group $G$ with Borel subgroup $B$, the above relations describe the algebra of functions on $G(\mathbb{F}_q)$ which are bi-invariant with respect to the left and right actions of $B(\mathbb{F}_q)$ under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group $G$, and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing $G$ and $B$ with $\text{SL}_2(\mathbb{Q})$ and $\text{SL}_2(\mathbb{Z})$ gives the Hecke operators in the theory of modular forms.

A (maximal) flag in a vector space $V$ of dimension $n$ is a chain $V_0 \subset V_1 \subset ... \subset V_n$ of subspaces such that $\dim V_i = i$. The flag variety of $G = \text{SL}(V) = \text{SL}_n$ is, for our purposes, the “space” of all maximal flags. $\text{SL}_n$ acts on the flag variety in the obvious way, and the stabilizer of any particular flag is a Borel subgroup $B$. If $e_1, ... e_n$ denotes a choice of ordered basis, one can define the standard flag $0, \text{span}(e_1), \text{span}(e_1, e_2), ...$, whose stabilizer is the space of upper triangular matrices of determinant $1$ with respect to the basis $e_i$. This is the standard Borel, and all other Borel subgroups are conjugate to it. Indeed, it’s not hard to see that $\text{SL}_n$ acts transitively on the flag variety, so the flag variety can be identified with the homogeneous space $G/B$.