Consider first passage Ising percolation on $Z^2$. Let $\beta$
denote the reciprocal temperature and let $h$ denote an external magnetic
field. Denote by $\beta_c$ the critical temperature and, for
$\beta\<\beta_c$, let
h_c(\beta) = h_c = \sup\{h:\theta(\beta, h) = 0\},
where $\theta(\beta,h)$ is the probability that the origin is
connected by an infinite (+)-cluster. With these definitions let us consider
first passage Ising percolation on $Z^2$. Let $a_{0,n}$ denote the first
passage time from $(0,0)$ to $(n,0)$. It follows from a subadditive argument
that
\lim_{n \to \infty} \frac{a_{0, n}}{n} = \nu \text{ a.s. and in } L_1.
It is known that $\nu > 0$ if $\beta < \beta_c$ and
$|h| < h_c(\beta)$. Here we will estimate the speed of the convergence,
\nu n \leq Ea_{0,n} \leq \nu n + C(n \log^5 n)^{1/2}
for some constant $C$. Define $\mu_{\beta,h}$ to be the unique
Gibbs measure for $\beta<\beta_c$. We also prove that there exist
$\tilde{C},\tilde{\alpha}>0$ such that
\mu_{\beta, h}(|a_{0,n} - Ea_{0,n} \geq x) \leq \tilde{C}
\exp(\\tilde{a}\frac{x^2}{n \log^4 n})
In addition to $a_{0,n)$, we shall also discuss other passage
times.