  As general theories, currently there are concentration inequalities (of
random walk) only for the cases of independence and martingale differences. In
this paper, the concentration inequalities are extended to more general
situations. In terms of the theory presented in the paper, the condition of
independence is $\frac { \partial y } {\partial t}=$ constant and martingale
difference's is $\frac { \partial y } {\partial t} = 0 $. This paper relaxes
these conditions to $\frac { \partial^2 y } {\partial u_i \partial t} \le L$;
i.e. $\frac { \partial y } {\partial t}$ can vary. Further, the concentration
inequalities are extended to branching random walk, the applications of which
solve some long standing open problems, including the well known problems of
K-SAT and K-COL phase transition locations, among others.
