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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 ...
2 votes
1 answer
165 views

Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?

A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
3 votes
0 answers
516 views

The distribution of collision stopping time in 2D random walk

Assume two particles A at $(0, 0)$ and B at $(a, b)$ in 2D discrete grid, both of them have the same possibility of $\frac{1}{4}$ for moving up/down/left/right (i.e. 2D random walk). We define the ...
1 vote
0 answers
663 views

The distribution of hitting time in 2D-lattice random walk [closed]

Assume a particle at $(0,0)$ with the same possibility of $1/4$ for moving up/down/left/right (i.e. random walk in 2D lattice). We define the stopping time 𝑇𝑐 as it hits $(a,b)$. How can we get the ...
57 votes
4 answers
15k views

Connectivity of the Erdős–Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
4 votes
1 answer
322 views

Approximating binomial coefficient sum

I have the following exact sum for the expectation of an event $$\sum_{m=0}^{nk} \sum_{j=1}^n (-1)^{j-1}\binom{n}{j} \binom{(n-j)k}{m} / \binom{nk}{m}$$ which is exactly correct but I want to give an ...
12 votes
3 answers
911 views

Expected number of compositions needed to get constant function

This is somewhat inspired by Factoring a function from a finite set to itself. Fix natural number $n$ and let $[n] := \{1,2,\ldots,n\}$. Set $g_0 \colon [n]\to [n]$ to be the identity, and for $i \geq ...
4 votes
1 answer
262 views

What is the number of finite Dynkin systems?

(This is a spin-off of Determine the minimal elements of a Dynkin system generated by a finite set of finite sets) Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power ...
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 ...
1 vote
1 answer
105 views

What is the distribution of a Cartesian power of a collection of iid uniform points? (renewed)

The following question was asked recently at https://mathoverflow.net/questions/326631/what-is-the-distribution-of-a-cartesian-power-of-a-collection-of-iid-uniform-poi : Take a rectangle with ...
4 votes
3 answers
324 views

Probability that $k$ random subsets of a fixed size covers a set

Let $A=\{1,\ldots,n\}$. Now, we uniformly randomly select $k$ subsets, $A_i$ of size $d$ from $A$. What is the probability that $\bigcup_i A_i=A$? This seems to be natural variant of the set cover ...
3 votes
3 answers
942 views

implementations of domino shuffling algorithm

Are there many implementations of the "domino shuffling" algorithm as found in William Jockusch, James Propp amd Peter Shor's Random Domino Tilings and the Arctic Circle Theorem math.CO/...
3 votes
1 answer
271 views

A quantity associated to a probability measure space

Let $(S,P)$ be a (finite) probability space. We associate to $(S,P)$ a quantity $n(S,P)$ as follows: The probability of two randomly chosen events $A,B\subset S$ being independent is denoted by $n(S,P)...
4 votes
0 answers
182 views

Determine the minimal elements of a Dynkin system generated by a finite set of finite sets

(This is a refined version of https://cs.stackexchange.com/q/144371) Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power set of $\Omega$ containing $\Omega$, which is ...
2 votes
1 answer
158 views

"Shape"/"norm" of a uniformly random set partition

Let $\mathcal{A}=\{A_1, A_2, \ldots, A_m\}$ be a uniformly random set partition of $[n]$. What can we say about $||\mathcal{A}||_2 = \sqrt{\sum_{i=1}^m |A_i|^2}$? It is clearly upper bounded by $n$, ...
1 vote
1 answer
125 views

Empirical degree distribution of random $n$ vertices labeled rooted tree converges to Poisson distribution

I am reading Louigi's lecture note on random trees and graphs here. I get stuck on part (b), Exercise 1.2.3 on page 19, which says the following: Let $T_n$ be uniformly drawn from $\mathcal{T}_n$, ...
5 votes
2 answers
707 views

Distribution of some sums modulo p

Fix a finite set of integers $S$ and a prime number $p$. Let $(a_1, a_2, \dotsc, a_n)$, $(b_1, b_2, b_3, \dotsc, b_n)$ be two sequences of integers where the numbers $a_i$ and $b_i$ are chosen ...
8 votes
1 answer
380 views

Question about estimating random symmetric sums modulo p

Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
2 votes
1 answer
199 views

Do enough permutations of an initial set probably cover most permutations?

Fix $\alpha, \epsilon \in(0,1)$. Take $(S_n)_n$ to be any sequence of sets with each $S_n$ containing $ \lceil (n!)^\alpha\rceil$ permutations of $n$ elements. Also build another sequence of sets $(...
3 votes
1 answer
118 views

Using singularity analysis for probability at a threshold?

In some urn model with parameter $p$, the generating function $$ f_p(z) \;=\; \frac{1+p\,z}{1-(1-p)\,z\,(1+p\,z)} $$ is such that $[z^n]f_p(z)$ is the probability that an $n$-urn configuration has a ...
3 votes
1 answer
241 views

Probability that k randomly drawn permutations can be arranged to compose to the identity

Consider the symmetric group $S_n$ under the uniform distribution. For integer $k > 1$, suppose we draw $k$ elements $s_1, \dots, s_k$ independently at random. What is the probability that there ...
11 votes
2 answers
1k views

Heuristic lower bounds on small sums of roots of unity

Let $f(k,n)$ be the smallest non-zero absolute value of a sum of $k$ complex $n$th roots of unity. Asking for bounds in either direction, Tao suggested that a polynomial lower bound seemed plausible ...
0 votes
1 answer
217 views

On independence of multiples of $\mathbb Z_p$

This is a rewording in combinatorial language of a question posed on another forum. The original was posed as a probabilistic problem. Problem set up: Consider for a fixed prime $p$, the ...
3 votes
1 answer
161 views

Probability permutation in turned to cycle

Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix). If you permute the columns and rows independently what is the probability resulting permutation matrix is a ...
1 vote
0 answers
47 views

Probability that a modified $c$-bounded lattice walk is also $c$-bounded

Let $\mathcal{B}_n^{(c)}$ be the set of $n$ step lattice bridges (starts at $(0,0)$ ends at $(n,0)$), taking steps $\{(1,1), (-1,1)\}$ bounded between $y = c$ and $y = -c$ for a constant $c \geq 0$. ...
13 votes
2 answers
518 views

Asymptotics of a randomized Fibonacci sequence

Let $f(1)=f(2)=1$ and recursively define $f(n+1) = f(n) + f(i)$, where $i$ is chosen uniformly at random from $1,\ldots,n-1$. About how big should we expect $f(n)$ to be for $n$ large? We can examine ...
1 vote
1 answer
119 views

Does exponential degree distribution entail Log-normal distance distribution in large complex graphs?

We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct ...
4 votes
1 answer
264 views

Information for discovering an item-colour assignment in a combinatorial game

We are given a set $S=\{i_1, i_2, \ldots, i_n\}$ of items and a set $C=\{c_1, c_2, \ldots, c_m\}$ of colours. Each item in $S$ is tinted with one colour $c\in C$. Let $\mathcal{A}$ be the set of all ...
1 vote
0 answers
176 views

Gaussian order statistics

Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one). Suppose $X_1,\dots,X_n$ are i.i.d. standard normal. Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
5 votes
1 answer
325 views

Bounding the max-loaded bin using${m \choose k} \|A\|_k^k$

Throw $m$ balls into $n$ bins independently, each ball selecting a bin from the distribution $A \in \Delta_n$. This question is about lower-bounding the max-loaded bin. Background. In this MO answer I ...
4 votes
0 answers
187 views

Dyadic distribution of $0/1$ permanents

Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$. What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...
2 votes
1 answer
843 views

Interpretation of probability statements in Nina Zubrilina's paper

I asked this question on Math.stackexchange but got no answer. In the paper Zubrilina - Asymptotic behavior of the edge metric dimension of the random graph (MR, the main result is $$\operatorname{...
1 vote
1 answer
173 views

Could you provide some TSP examples from real world to test a new algorithm?

It's well known that to find a hamilton cycle is NPC, while TSP is NPH. But it seems that for majority of graphs (density of edge > 0.1, order > 100) there is a fast algorithm to find different ...
1 vote
1 answer
207 views

Expectation of the sum of the squares of the cardinal of an inverse function

I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as: $$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$ where $\oplus$ is the bitwise XOR. ...
3 votes
1 answer
128 views

A ratio of two probabilities

I am concerned about the monotonicity of the following ratio $ f(\eta)=\frac{\sum_{x=K}^{N}\left(\begin{array}{c} N\\ x \end{array}\right)\left(q_{G}\eta\right)^{x}\left(1-q_{G}\eta\right)^{N-x}}{\...
1 vote
1 answer
141 views

Expectation of maximum of all period lengths of functions $f:\{1,\ldots,n\}\to\{1,\ldots,n\}$

This is based on an older question. For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ and ''starting value'' $...
6 votes
2 answers
274 views

Expectation of period length of functions $f:\{1,\ldots,n\}\to \{1,\ldots,n\}$

For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ associate a sequence $\text{seq}(f))$ defined recursively by ...
5 votes
3 answers
601 views

Convergence speed of a random dyadic rational generator

We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$ two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
1 vote
1 answer
106 views

Almost-parallel corners of the hypercube in high dimensions

Say I would like a collection of k "almost-parallel" boolean vectors $X_1,...,X_k \in \{\pm 1\}^n$, such that $(X_i,X_j)/n \approx 1-\epsilon$ for some small $\epsilon$. How many ways are ...
6 votes
0 answers
156 views

Distribution of iid hypergeometric random variables conditioned on the sum

Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$....
3 votes
1 answer
206 views

Random planes separating points in $\mathbb{R}^3$

We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
1 vote
0 answers
127 views

Delocalization of eigenvectors of graph Laplacians

Let $(V,E)$ be an undirected, connected graph with $n$ nodes. The graph Laplacian is defined as $L = D - A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. Let $0 = \lambda_1 < \...
3 votes
1 answer
153 views

Randomized version of Turán's theorem II

$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then \begin{equation} \om(G)\ge\...
5 votes
1 answer
209 views

Randomized version of Turán's theorem

Turán's theorem says the following. Take any natural $n$ and $r$. Suppose that \begin{equation*} |G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0} \end{equation*} where $|G|$ is the number of edges of ...
0 votes
0 answers
133 views

is there an example in planar graph that using probabilistic methods

The probabilistic method is a technique for proving the existence of an object with certain properties by showing that a random object chosen from an appropriate probability distribution has the ...
8 votes
2 answers
475 views

Random permutations without double rises (avoiding consecutive pattern $\underline{123}$)

A permutation avoiding a consecutive pattern $\underline{123}$ is permutation $\pi = \pi_1 \pi_2 \ldots \pi_n$ with the property that there does not exists $i \in [1, n-2]$ such that $\pi_i < \pi_{...
2 votes
1 answer
426 views

Random subgraph properties

Consider a graph $G$ of $N$ vertices and $M$ edges, and assume $G$ has typical complex network properties: it is not necessarily connected, but it has a high clustering coefficient and a giant ...
2 votes
1 answer
151 views

Given an input point in $\mathbb{R}^n$, select (one of) the closest point(s) from a fixed large set of points given in advance

We are given a set $S$ of $m\gg 1$ points in $\mathbb{R}^n$. In the problem I am trying to solve, in a sequential fashion, we obtain a new point $p_r\not\in S$ at each round $r\ge 1$ and the goal is ...
6 votes
0 answers
321 views

extensions of the Nekrasov-Okounkov formula

This post is related to the issues addressed in A q,t-extension of Plancherel Measure thru Yang-Mills Theory ? however the generalization/interpolation which John Mangual asks for looks different ...
3 votes
1 answer
315 views

Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)

If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...

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