This paper focuses on the time constant for last passage percolation on complete graph. Let Gn=([n],En) be the complete graph on vertex set [n]={1,2,…,n}
Nearby in the stack
, and i.i.d. sequence
{Xe:e∈En}
be the passage times of edges. Denote by
Wn
the largest passage time among all self-avoiding paths from 1 to
n
. First, it is proved that
Wn/n
converges to constant
μ
, where
μ
is called the time constant and coincides with the essential supremum of
Xe
. Second, when
μ<∞
, it is proved that the deviation probability
P(Wn/n≤μ−x)
decays as fast as
e−Θ(n2)
, and as a corollary, an upper bound for the variance of