All Questions
Tagged with gr.group-theory pr.probability
27 questions with no upvoted or accepted answers
21
votes
0
answers
578
views
Density of first-order definable sets in a directed union of finite groups
This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \...
- 2,200
11
votes
0
answers
263
views
Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?
Let $G$ be a finite group. It has been shown that:
If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian.
If the probability ...
- 495
10
votes
0
answers
3k
views
Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog
I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-...
- 6,034
10
votes
0
answers
809
views
Where can I find analogues of combinatorial central limit theorems for other groups
The statement of Hoeffding's combinatorial central limit theorem is as follows: given for each $n$, an $n \times n$ matrix $A = (a_{ij})$, one can consider the random diagonal sum:
$$\displaystyle f(\...
- 4,466
8
votes
0
answers
211
views
Superharmonic functions and amenability
Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$.
Assume that there is a set of non-...
- 4,705
7
votes
0
answers
233
views
Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?
Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
- 24.9k
7
votes
0
answers
743
views
Distribution of the sizes of conjugacy classes in the symmetric group.
This recent question makes me wonder: is there some known limit theorem for the distribution of the sizes of conjugacy classes in the symmetric group $S_n?$ A quick search seems to reveal nothing ...
- 96.4k
6
votes
0
answers
301
views
Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
- 8,512
6
votes
0
answers
295
views
Is there an idempotent measure on the free LD system?
This is a follow up question to MO question "Idempotent measures on the free binary system?".
Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law:
...
- 3,547
4
votes
1
answer
223
views
Existence of disintegrations for improper priors on locally-compact groups
In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
- 8,512
4
votes
0
answers
266
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 ...
- 24.9k
4
votes
0
answers
177
views
Is there any probabilistic characterization for generalized solvable groups?
References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers:
(1) On the probability of satisfying a word in ...
user82740
3
votes
0
answers
153
views
Metropolis-Hastings sampling as a group action
Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...
2
votes
0
answers
100
views
Distributions of random walks on boundaries of balls in hyperbolic metric spaces
Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...
- 21
2
votes
1
answer
150
views
Can we find background noise for every Følner sequence in a countable amenable group?
Let $G$ be a countable amenable group. We consider sequences $(z_g)_{g\in G}$ of complex numbers with $|z_g|=1$ for all $g\in G$.
I will say $(z_g)_{g\in G}$ is background noise for a (left-)Følner ...
- 10.6k
2
votes
0
answers
138
views
Update on Viskov's paper on random processes, Lagrange inversion, and the Heisenberg–Weyl algebra
"A Random Walk with a Skip-Free Component and the Lagrange Inversion Formula" by Viskov presents connections among Lagrange inversion and measures of random Lévy processes. The freely ...
- 10.5k
2
votes
0
answers
110
views
Moment of the hitting measure of a subgroup
Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
- 4,432
2
votes
0
answers
89
views
Name for a probability density ''symmetrized'' by a permutation group?
Let $p$ be a probability density function over random variable $X$, and $G$ a compact permutation group over the outcomes of $X$. For each $g\in G$, let $p_g$ indicate the probability density ...
- 695
2
votes
0
answers
202
views
Random walk on a finite group, converging modulo a function
Let $G$ be a finite group, and let $Q$ be a probability measure on $G$. Suppose that $Q$, as a function on $G$, is supported on a conjugacy class $C$. We denote by $Q^{*k}$ the $k$-fold convolution of ...
- 14.6k
2
votes
0
answers
88
views
Example of action of an infinitely countable group that has important ergodic/statistical property?
I work in probability and I am looking for an important example of action of an amenable countable group in other areas of math for which the (pointwise) ergodic theorem is actually quite important. ...
- 21
2
votes
0
answers
153
views
Reference request for a result on subsets unlikely to be hit by random walks in a group
Suppose we are performing a random walk in a group. More precisely, we have a finite generating set $S$ of a group $G$ and the probability of walking along generator $s$ is given by $\mu(s)$ for some ...
- 21
1
vote
0
answers
177
views
Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
vote
1
answer
185
views
A system of linear equations with way too many unknowns — constructing a bivariate distribution from marginals and "the diagonal"
Suppose we are given information about distributions of random permutations $\sigma, \tau : \Omega \to S_n$ as follows:
$$p^1_{k,l} = \mathbb P(\sigma(k) = l), p^2_{k',l'} = \mathbb P(\tau(k) = l), p^{...
- 677
1
vote
0
answers
489
views
Can we generalize the concept of "characters" in group theory via methods from statistics and probability theory?
$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random ...
- 356
1
vote
0
answers
340
views
Random walk on non-abelian free group
Let $F_2$ be the free non-abelian group with generators $a, b\in F_2$.
Has the "random walk" where we start with the identity and then multiply it by $a$ or $b$ or $a^{-1}$ or $b^{-1}$ ...
- 11
1
vote
0
answers
311
views
Show that $\mathrm{SL}_2(\mathbb{F}_p)$ is quasi-random
Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at least $...
- 22.8k
0
votes
0
answers
118
views
A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101