Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on the vacant set of random walks on the hypercube and other
  regular graphs of high degree   We consider a random walk on a $d$-regular graph $G$ where $d\to\infty$ and
$G$ satisfies certain conditions. Our prime example is the $d$-dimensional
hypercube, which has $n=2^d$ vertices. We explore the likely component
structure of the vacant set, i.e. the set of unvisited vertices. Let
$\Lambda(t)$ be the subgraph induced by the vacant set of the walk at step $t$.
We show that if certain conditions are satisfied then the graph $\Lambda(t)$
undergoes a phase transition at around $t^*=n\log_ed$. Our results are that if
$t\leq(1-\epsilon)t^*$ then w.h.p. as the number vertices $n\to\infty$, the
size $L_1(t)$ of the largest component satisfies $L_1\gg\log n$ whereas if
$t\geq(1+\e)t^*$ then $L_1(t)=o(\log n)$.

11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an