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231 votes
13 answers
42k views

Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
Pete L. Clark's user avatar
9 votes
1 answer
889 views

Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory

In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
Tom Copeland's user avatar
  • 10.5k
35 votes
5 answers
11k views

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
user4's user avatar
  • 921
62 votes
7 answers
10k views

Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
Randy Qian's user avatar
430 votes
16 answers
65k views

Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
Andrej Bauer's user avatar
  • 48.8k
91 votes
13 answers
146k views

If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...
Michael Lugo's user avatar
63 votes
3 answers
7k views

A roadmap to Hairer's theory for taming infinities

Background Martin Hairer gave recently some beautiful lectures in Israel on "taming infinities," namely on finding a mathematical theory that supports the highly successful computations from quantum ...
Gil Kalai's user avatar
  • 24.7k
42 votes
3 answers
5k views

The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
Denis Serre's user avatar
  • 52.3k
27 votes
5 answers
3k views

Nice applications for Schwartz distributions

I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are: Some multilinear algebra including the Kernel Theorem and ...
Abdelmalek Abdesselam's user avatar
7 votes
0 answers
579 views

Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
Tom Copeland's user avatar
  • 10.5k
40 votes
5 answers
6k views

Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question): The Riddle: We assume ...
Denis's user avatar
  • 1,341
26 votes
5 answers
10k views

Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...
Michael Lugo's user avatar
7 votes
1 answer
880 views

Bound for largest eigenvalue of symmetric matrices of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s

Given a $n\times n$ symmetric random matrix whose diagonal elements are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding ...
Tony's user avatar
  • 272
5 votes
1 answer
392 views

Uniqueness of the solution to some SDE

Consider the stochastic differential equation as follows: $$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$ where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
GJC20's user avatar
  • 1,334
222 votes
0 answers
18k views

Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
Sil's user avatar
  • 2,272
74 votes
16 answers
8k views

Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
60 votes
1 answer
7k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...
Benjamin Dickman's user avatar
52 votes
4 answers
13k views

Central limit theorem via maximal entropy

Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that: $$\int_\mathbb{R} \rho(x)\, dx = 1$$ and $$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$ ...
Paul Siegel's user avatar
  • 29.2k
41 votes
6 answers
4k views

Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative ...
Joseph O'Rourke's user avatar
25 votes
1 answer
4k views

What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $...
Qiaochu Yuan's user avatar
21 votes
7 answers
14k views

A balls-and-colours problem

A box contains n balls coloured 1 to n. Each time you pick two balls from the bin - the first ball and the second ball, both uniformly at random and you paint the second ball with the colour of the ...
Hedonist's user avatar
  • 1,269
18 votes
1 answer
890 views

Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...
Simd's user avatar
  • 3,377
10 votes
4 answers
9k views

Mean minimum distance for N random points on a unit square (plane)

A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random ...
KEN KEL's user avatar
  • 111
10 votes
3 answers
4k views

Expected supremum of average?

Is there either a closed form (in terms of the moments of $X_1$, say) or good bounds on $$ \mathbb{E} \sup_{k \leq n} \frac{1}{k} \sum_{i=1}^k X_i, $$ where $X_i$ are iid and arbitrarily nice? (In my ...
Elena Yudovina's user avatar
10 votes
2 answers
925 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
Abdelmalek Abdesselam's user avatar
10 votes
2 answers
1k views

Why Kleisli Markov categories and not the Eilenberg-Moore categories of the associated monads

Why is there so much interest in the Markov categories which are Kleisli categories for monads corresponding to distributions etc. but not much discussion of the E.M. categories? For example, the E.M. ...
Q.Faustus's user avatar
  • 101
8 votes
2 answers
3k views

Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?

Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$. It happens that the ...
Linus Hamilton's user avatar
7 votes
3 answers
2k views

Packing density of randomly deposited circles on a plane

Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$: Under "no overlap" constraints, I sequentially deposit circles of radii $r_c$ on this surface,...
user14324's user avatar
  • 309
6 votes
1 answer
998 views

A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
JustWannaKnow's user avatar
6 votes
1 answer
684 views

Differentiable dependence on the initial condition of the solution of a SDE

Let $b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-...
0xbadf00d's user avatar
  • 167
6 votes
1 answer
705 views

Reformulation - Construction of thermodynamic limit for GFF

I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
JustWannaKnow's user avatar
4 votes
0 answers
264 views

Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
Alexander Chervov's user avatar
2 votes
1 answer
675 views

Distribution of ratio between complex Gaussian and Chi-square R.V.s

What would be the distribution (p.d.f.) of the following ratio? $$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$ where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can ...
Felipe Augusto de Figueiredo's user avatar
191 votes
34 answers
81k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
106 votes
5 answers
10k views

integral of a "sin-omial" coefficients=binomial

I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof? For any pair of integers $n\geq k\geq0$, we have $$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
T. Amdeberhan's user avatar
72 votes
9 answers
30k views

When are probability distributions completely determined by their moments?

If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. ...
Steve Flammia's user avatar
52 votes
5 answers
2k views

Tetris-like falling sticky disks

Suppose unit-radius disks fall vertically from $y=+\infty$, one by one, and create a random jumble of disks above the $x$-axis. When a falling disk hits another, it stops and sticks there. Otherwise, ...
Joseph O'Rourke's user avatar
44 votes
5 answers
7k views

Heuristically false conjectures

I was very surprised when I first encountered the Mertens conjecture. Define $$ M(n) = \sum_{k=1}^n \mu(k) $$ The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...
36 votes
2 answers
13k views

Mean minimum distance for N random points on a one-dimensional line

Let's say that I have a one-dimensional line of finite length 'L' that I populate with a set of 'N' random points. I was wondering if there was a simple/straightforward method (not involving long ...
Mensen's user avatar
  • 811
25 votes
6 answers
10k views

Metrization of weak convergence of signed measures

Edit: Changed from "Hausdorff" to "metric" spaces. Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, ...
Dirk's user avatar
  • 12.7k
25 votes
6 answers
6k views

Proof of Krylov-Bogoliubov theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
Quinn Culver's user avatar
24 votes
6 answers
3k views

Shortest grid-graph paths with random diagonal shortcuts

Suppose you have a network of edges connecting each integer lattice point in the 2D square grid $[0,n]^2$ to each of its (at most) four neighbors, {N,S,E,W}. Within each of the $n^2$ unit cells of ...
Joseph O'Rourke's user avatar
22 votes
2 answers
2k views

Can one view the Independent Product in Probability categorially?

One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a ...
Michael Greinecker's user avatar
20 votes
2 answers
6k views

Constants in the Rosenthal inequality

Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $X = \sum_{i=1}^n X_i$. Then we have the family of "Rosenthal-type ...
Jelani Nelson's user avatar
20 votes
1 answer
2k views

Does every compact metric space have a canonical probability measure?

Edit: Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is ...
M. Kelly's user avatar
  • 203
18 votes
3 answers
8k views

Number of invertible {0,1} real matrices?

This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$. My question is: how many such matrices ...
Tony Huynh's user avatar
  • 32.1k
18 votes
3 answers
1k views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
Joseph O'Rourke's user avatar
17 votes
4 answers
2k views

Good introduction to statistics from a algebraic point of view?

There are already lots of questions on this subject like Is there an introduction to probability theory from a structuralist/categorical perspective? Is there a combinatorial/topological treatment ...
doofin's user avatar
  • 283
17 votes
1 answer
732 views

Reference request: a conjecture of Rota on positive functions of a random variable

Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows: Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2, ...
Qiaochu Yuan's user avatar
16 votes
3 answers
2k views

Integration of a function over 7-sphere

Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$. The problem is finding or approximating the ...
Hrushikesh Pawar's user avatar

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