All Questions
Tagged with pr.probability gr.group-theory
102 questions
7
votes
3
answers
801
views
Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II
For some context see Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance
As per Noah's answer and JBL's comment this was false as stated. However, I think the following ...
3
votes
1
answer
3k
views
variance of the number of fixed points for a permutation group
It is reasonably well-known that the variance of the number of fixed points for $S_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I ...
- 96.4k
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
3
votes
1
answer
693
views
Size of automorphism group of random regular graph
If I pick a random regular graph on $n$-vertices and degree $d$ from uniform distribution what is the probability that its automorphism group is of size at least $m$?
--
I want to know what is the ...
- 13.9k
6
votes
1
answer
569
views
Liouville property - a very basic question
Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...
- 4,705
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
7
votes
2
answers
1k
views
An Expectation of Cohen-Lenstra Measure
The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to ...
- 22.8k
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
8
votes
2
answers
755
views
The Odds 3 (or More) Group Elements Commute
Some time ago I asked about the odds 2 group elements commute. I wonder about the odds that 3 group elements commute. Is there a "closed" formula for the sum
$$ \frac{1}{|G|^3} \sum_{g,h,k} \delta([...
- 22.8k
15
votes
3
answers
3k
views
Entropy of a measure
Let $\mu$ be a probability measure on a set of $n$ elements and let $p_i$ be the measure of the $i$-th element. Its Shannon entropy is defined by
$$
E(\mu)=-\sum_{i=1}^np_i\log(p_i)
$$
with the ...
- 6,034
1
vote
2
answers
635
views
Mean value theorems for the Haar integral?
Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems ...
- 5,407
5
votes
3
answers
314
views
Tracking automorphism groups of graph processes
Start with an edgeless graph on $n$ labeled vertices, and note that the automorphism group is $\Sigma_n$, the symmetric group on $n$ elements. Now imagine that we randomly start throwing in all of the ...
- 15.5k
2
votes
2
answers
381
views
Speed and absence of non-constant bounded harmonic functions
For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...
- 4,432
9
votes
3
answers
654
views
measure with given push-forwards
Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps $p_1,\ldots,p_n:Y\...
12
votes
3
answers
552
views
Estimate on currents in Cayley graphs
Take a Cayley graph $\Gamma$ (thought of as an electrical network with all edges having equal resistance) and break one edge $e$ and put a battery there. (Assume the graph has only one end* so that ...
- 4,432
9
votes
2
answers
726
views
Return probabilities for random walks on infinite Schreier graphs
Question: Is there a sequence $(\delta_n)_n$ of real numbers with $\delta_n \to 0$ as $n \to \infty$, such that the following holds:
Let $F$ be a free group on two generators, let $F \curvearrowright ...
- 25.5k
5
votes
1
answer
774
views
Probabilities of a random walk exiting a set
Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as ...
- 4,432
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
23
votes
3
answers
1k
views
In an inductive family of groups, does the probability that a particular word is satisfied converge?
We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...
- 5,898
7
votes
2
answers
639
views
Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
18
votes
1
answer
996
views
Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
8
votes
1
answer
452
views
What is the probability that a random subset of a finite group is generic?
Definition 1: Given a group $G$, a subset $X \subseteq G$, and a natural number $k$,
we say that $X$ is (left) $k$-generic in $G$ if there are $k$ many left translates of $X$ that cover $G$.
That is, ...
- 83
2
votes
1
answer
261
views
Does the set of automorphisms of a cyclic group exhibit some sense of randomness?
I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome.
...
- 2,480
4
votes
1
answer
400
views
Speed of random walks in groups
I've seen some estimates for the decay in $d$ of the probability a SRW makes a distance $d$ in time $n$, but is there any reference for the "speed" of a random walk in a group? I'm interested mostly ...
- 4,432
8
votes
3
answers
606
views
Many Brownian motions moving together
Let $ (B^i),\:{{i=1,\ldots,n}}$ be a set of independent Brownian motions. By $(X^i)$ we denote $(B^i)$ conditioned on the event
$|B^i_t-B_t^{i+1}|\leq 1,\quad \forall_{1\leq i\leq n-1}, \forall_{t\...
- 1,491
4
votes
1
answer
557
views
Symmetries of the standard probability space
The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In applications,...
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
6
votes
1
answer
844
views
Random geometries
Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its tangent frame bundle. Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all $G$-...
- 8,512
4
votes
1
answer
204
views
Estimates for simple random walks in groups of intermediate growth
I'm looking for references for the rate of escape and return probability for a group of intermediate growth.
Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then (...
- 4,432
6
votes
2
answers
461
views
Intrinsically measurable subsets of amenable semigroups.
This question is related to the one in https://mathoverflow.net/questions/65322/the-structure-of-certain-maximal-sets-of-means-into-amenable-semigroups. I open a different topic because they can be ...
- 6,034
5
votes
1
answer
437
views
Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
- 8,512
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
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
1
vote
1
answer
578
views
Fundamental inequality of entropy in random walks
I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,
$\textbf{Question}$: ...
- 4,432
3
votes
1
answer
324
views
A stronger version of supramenability?
A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger ...
- 2,555
2
votes
1
answer
173
views
Proving that a closed walk of some odd length k exists on a Cayley graph
I'm trying to prove the following:
Let $G$ be a group with finite symmetric generating set $S$ and let $\Gamma(G,S)$ be the corresponding Cayley graph. Let $X_1, X_2,\cdots$ be a simple random walk ...
- 179
3
votes
1
answer
539
views
Probability of generating the symmetric group
The statement is simple:
What is the probability that a set of n-1 transpositions generates the symmetric group, $S_n$?
The motivation is that I remembered reading that this was an open problem ...
- 543
0
votes
3
answers
293
views
How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$?
Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s_1,...s_k\in S$ such that the sets $s_i\cdot W$ are pairwise ...
- 6,034
-2
votes
1
answer
190
views
Dixon's Theorem [closed]
I am going through a sketch of the proof of Dixon's Theorem (the probability that two randomly chosen elements of A_n generate A_n -> 1 as n -> infinity) due to M. Liebeck and its underlying idea is ...
- 65
3
votes
1
answer
263
views
Extending Tarski's Theorem on invariant measures
Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$.
I am ...
- 2,555
4
votes
2
answers
420
views
Generating a group by randomly sampling generators
Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidentity element of $G$ in ...
- 2,649
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
2
votes
1
answer
337
views
Genericity of sets without unique mean value
Following Rosenblatt and Yang, I say that a subset $A$ of $\mathbb Z$ has a unique mean value if for all invariant means $\lambda_1,\lambda_2$ on $\mathbb Z$, one has $\lambda_1(A)=\lambda_2(A)$.
...
- 6,034
2
votes
1
answer
141
views
Spanning subgaph with trivial Poisson boundaries
Assume $\Gamma$ is the Cayley graph of an amenable$^{*}$ group and that the simple random walk has non-trivial Poisson boundary$^{**}$. Is there a spanning connected subgraph $\Gamma'$ of some $k$-...
- 4,432
2
votes
2
answers
389
views
Who is the weak* sequential closure of the set of finitely supported measures on the integers?
Let $X$ be a topological space and $Y\subseteq X$, the sequential closure of $Y$ is the set of elements in $X$ that are limit of sequences belonging to $Y$.
Let $\mathcal M_{\text{fin}}(\mathbb Z)$ ...
- 6,034
9
votes
1
answer
526
views
Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance
Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start ...
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
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
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
1
vote
1
answer
259
views
Amenability with respect to a function
Let $(G,\cdot)$ be a group and $\phi:G\rightarrow\mathbb R$ bounded. Let me say that the pair $(G,\phi)$ is amenable if there is a finitely additive probability measure $\mu$ on $G$ such that for all $...
- 6,034