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

Solutions to the Problem of K-SAT / K-COL Phase Transition Location