# Tagged Questions

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### Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...
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### Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots, S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin. Let $\tau_{N}$ be the first time $S_{n}$ exits ...
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### Time for Langton's ant to cover a “square” torus

Langton's ant is a cellular automaton running as follows: Squares on a plane are colored variously either black or white. We arbitrarily identify one square as the "ant". The ant can travel ...
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### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks on $\mathbb{Z}^2$ that turn left or right at each step (i.e., they cannot continue the direction of the preceding step) have been ...
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### How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?

Background Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...
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### On “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)

I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...
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### Any approximation algorithms for self-avoiding walks?

I've a graph whose edges are weighted by probabilities, perhaps all equal. I would like to compute the overall probability of traveling between vertices x and y in the graph after I delete each edge ...
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### Compute the expected value of the next step of a sorted random walk

Here's what I'm thinking about. If you have a random walk (move +1 or -1 at each step) of some fixed length, then if you're at the maximum of the walk, the next step you take is -1 with probability 1. ...
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### Hierarchical (Recursive) Random Walk Model

Consider the following hierarchical (recursive) random walk model. For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete) random walk. For the next level $\ell=2$, we ...
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### Do the normal forms for braid groups really conceal information about better than randomly applying the braid relations?

In braid based cryptography, one typically wants to conceal the way a certain braid $b$ has been obtained. One therefore puts $b$ into some normal form. Since every braid has a unique normal form, the ...
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Let $X_t$ be a random process such that \begin{eqnarray} X_1 &=& 0\\ X_t &=& X_{t-1} + \left\{\begin{array}{ll} A_t, & X_{t-1} \geq 0\\ B_t, & X_{t-1} < 0 \end{array}\...
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### Why is the density function of a sum of many iid random variables (i.e., the Gaussian) self-dual?

In the usual proof of the central limit theorem via characteristic functions, we note that if $X_1, \dots, X_n$ are independent and identically distributed, with probability density function $f(x)$, ...
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### 1d random walk probability of previous n positions

I have the following question. (May be it is very simple, but I cannot find the answer). Suppose I have a 1d random walk on integer numbers with equal pobabilities of unit step in either direction (...
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### Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...
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### Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height

I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...
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### Random walk on the hypercube

Let $H_N=\{0,1\}^N$ the N-dimensional hypercube. We define the following random walk $X_n$ on $H_N$: start from a point $x \in H_N$ pick at random an integer $k$ in $[1,N-1]$ and exchange $x(k)$ and ...
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### Monotonic properties of harmonic functions on graphs

I have a question concerning monotonic properties of "generalized harmonic functions" on graphs. I am a physicist and I didn't take any separate courses in neither graph theory nor discrete harmonic ...
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### What is the influence of unreliable comparisons on the results of sorting

Considering sorting algorithms based solely on binary comparisons of the elements to be sorted(algorithms such as insertion sort, selection sort, quicksort, and so on), what problems do we face when ...
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### Is the family of probabilities generated by a random walk on a finitely generated amenable group asymptotically invariant?

Is the family of probabilities $\mu^n$ (convolution) generated by a random walk $\mu$ on a finitely generated amenable group $G$ asymptotically invariant ($\|g\mu^n-\mu\|_{L^1}\to 0$ for any $g\in G$)?...
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### Uniform sub-linearity of sub-additive functions on groups

Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...
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### criterion for deciding whether the product of a sequence of Givens rotations can reach the full special orthogonal group

By Givens' rotation $R(1,2;\theta)$ I mean a matrix which has the $$\begin{pmatrix} \hphantom{-}\cos \theta &\sin \theta \cr -\sin \theta & \cos \theta \end{pmatrix}$$ $2 \times 2$ block at ...
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### Controlling fluctuations in a Markov chain

For $N>0$, consider the Markov chain $x_n$ on $\{0,1/N,...,1\}$ that moves up by $1/N$ at rate $(c-x_n/2)N$ and down by $1/N$ at rate $(c+x_n/2)N$. As $N\rightarrow\infty$ sample paths approach ...
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### Multi-particle random walks

Define a multi-particle "breeding" random walk $\mathcal{W_p}$ in $d$ dimensions, for $p \in (0,1)$ as follows: The state of $\mathcal{W_p}$ at integer time $t\geq 0$ consists of the pair $(k, x)$ ...
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### Large-$t$ expected distance from origin in non-trapping, self-avoiding random walks

Consider two variants on Self-(vertex-)Avoiding Random Walks on $\Bbb{Z}^2$: (A) "Legal" steps consist of any step not ending on a vertex previously visited, and the probabilities of each of the 1, 2,...
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### Unidirectional continuous path discrete time random walk

Is there any material available to study on unidirectional continuous path discrete time random walk on a line interval. To say "unidirectional continuous path discrete time random walk on a line ...
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### Markov chains on a polyhedron

A modification of a question from Gerard Letac (1976): A m-sided q-adjacent-faced polyhedron has one of its faces "up." Each round, the polyhedron rolls so that any of the adjacent faces is now up. ...
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### Probability density function of the node positions in a random walk after N time slots

Hello, my question basically is how do I find the probability density function of the position of the nodes in a given area after N discrete time slots when the nodes move following the 2D random walk ...
scribd.com/doc/87930409/18/Leads-for-tied-down-random-walk [Theorem 3.7, page 38] Could you explain me the last equality in the proof? I mean this: \frac{2[\sqrt{1 - s^2t^2} - \sqrt{1-t^2}]}{t^2(...
I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...