# Questions tagged [random-walks]

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### A symmetry of lattice paths

The number of $n$-step NSEW lattice paths from $(0,0)$ to $(a,b)$ that intersect the line $y=k$ precisely $t$ times is independent of $k$, for $0\leq k\leq b$, where we assume $b\geq0$ for simplicity. ...
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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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### Is the P.M.F. of the first return time of a random walk monotone?

Suppose $X_1,X_2,\ldots$ are i.i.d. $\mathbb Z$-valued random variables such that the random walk $$S_n=\sum_{i=1}^nX_i$$ is recurrent with some period $k\geq1$ (i.e., $\Pr[S_n=0]>0$ if and only if ...
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### 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-...
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### Asymmetric random walk on the line with barriers

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers). More generally, suppose we fix any ...
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### Sum of variables uniformly distributed on a circle: a cyclic property

Consider a random walk starting at the origin in the plane, walking $n$ steps in independent uniformly random directions with step lengths $a_1,\ldots,a_n$, and observing the distance from the origin. ...
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### Random walks in arrangements of lines in the plane

Let $\cal{A}_n$ be a simple arrangement of $n$ lines in $\mathbb{R}^2$. (Simple: each pair of lines meet in a distinct point, i.e., no three lines pass through the same point.) Start a random walk at ...
163 views

### Random Up-walk on Young's Lattice

Starting from the empty partition, $\varnothing$, follow a random up-walk of length $n$ on Young's Lattice, where an edge's transition probability is 1/updegree. For a particular partition, $\lambda$, ...
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### Chains of right annihilators in group rings

See the update below This problem emanates from a question on not-so-simple random walks on finitely generated groups. But to explain the connections would require an extremely long essay. Let $G$ be ...
391 views

### Hierarchical Random Walk (also known as Hierarchical Hidden Markov Model)

Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...
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