Infinite Resistive lattice problem

While there have been many discussions regarding the solution(say using the Fourier transform or the lattice Green's function approach) to the two-point resistance of an infinite-square resistive lattice my doubt is actually related to one thing common to almost all the approaches to the problem i.e how does having applied a unit current and removed the same current between the points on which one wants to determine the effective resistance between, imply that the currents at the other nodes are zero.Moreover in order to obtain the difference equation (discretized Laplacian) one makes use of the Kirchoff's current law(which talks about the current in the branches connecting neighboring nodes).

So say i applied a unit current, $I_{m,n}=\delta_{m,mo}\delta_{n,no}- \delta_{m,0}\delta_{n,0}$ at the $(m,n)$th point on the lattice. In order to use my KCL i would need talk about the current in the branches which are incident at the point/node (m,n).So this means that the current emanates from point $A$ and reaches $B$, $C$, $D$ and $E$. But then as soon as it reaches those points does it become zero??

But this doesn't make sense right since we are removing the same unit current from another point(different from the m,nth point on the lattice) in this case from node $O$.

Could someone resolve the same for me.

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Related: physics.stackexchange.com/q/2072/2451 , physics.stackexchange.com/q/10308/2451 and links therein. – Qmechanic Apr 23 '14 at 21:44

" i.e how does having applied a unit current and removed the same current between the points on which one wants to determine the effective resistance between, imply that the currents at the other nodes are zero."

It doesn't. It implies that the net current at each of the other nodes is zero. In your example, node D, for instance, will have considerable current flowing through it, but the net current will be zero, since all the current flowing into the node also flows out of the node.

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If i am not wrong what you mean by the 'net current' is the time averaged current.So what your saying is that the time averaged current going through the nodes is zero.But the thing is that this problem isn't a dynamical problem in the sense that there are no inductors and capacitors in the lattice.So where does the time dependence come from?? – user42382 Mar 24 '14 at 16:26
Net current means current flowing into a point minus current flowing out of a point. If the net current wasn't zero, charge would build up there. The only place current can be non-zero is at the two points of interest. – lionelbrits Dec 21 '14 at 23:08
Current flowing point $A$ will reach point $B$ from four different directions, not just the direct route. And because charge can't build up anywhere, the sum of all the currents reaching $B$ has to be zero. – lionelbrits Dec 21 '14 at 23:10