Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase transition in a sequential assignment problem on graphs   We study the following game on a finite graph $G = (V, E)$. At the start,
each edge is assigned an integer $n_e \ge 0$, $n = \sum_{e \in E} n_e$. In
round $t$, $1 \le t \le n$, a uniformly random vertex $v \in V$ is chosen and
one of the edges $f$ incident with $v$ is selected by the player. The value
assigned to $f$ is then decreased by $1$. The player wins, if the configuration
$(0, \dots, 0)$ is reached; in other words, the edge values never go negative.
Our main result is that there is a phase transition: as $n \to \infty$, the
probability that the player wins approaches a constant $c_G > 0$ when $(n_e/n :
e \in E)$ converges to a point in the interior of a certain convex set
$\mathcal{R}_G$, and goes to $0$ exponentially when $(n_e/n : e \in E)$ is
bounded away from $\mathcal{R}_G$. We also obtain upper bounds in the
near-critical region, that is when $(n_e/n : e \in E)$ lies close to $\partial
\mathcal{R}_G$. We supply quantitative error bounds in our arguments.

31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat