How to do this path/lattice with tikz

I am looking for a way to do this lattice with tikz, and other similars to it.

There are other similars to it, any idea?

• I know how to do it using " brute strength", doing every line, arrow, dot in the diagram, but this is not usefull or handly, I dont know if is there an "intelligent" way. – J.Rodriguez Apr 16 at 2:33
• Add it to the question anyway. It is helpful to us – user156344 Apr 16 at 2:34

It is not so hard with decorations.markings:

\documentclass[tikz]{standalone}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}
\draw (0,0) grid (5,5);
\foreach \i in {0,...,5}
\foreach \j in {0,...,5}
\fill (\i,\j) circle (1pt);
\draw[ultra thick,red,postaction=decorate,decoration={
markings,
mark=between positions {0.5/11} and {10.5/11} step {1/11} with {
\draw[-stealth,very thin] (-.4,.1)--(.4,.1);}
}] (0,2) -- (2,2) |- (1,1) |- (4,5);
\fill[red] (0,2) circle (2pt) node[left] {Identity}
(4,5) circle (2pt) node[above] {End of path};
\end{tikzpicture}
\end{document}