In the [Lindström–Gessel–Viennot lemma][1] (LGV) applied the $Z^2$ lattice paths are taken to move in unit spatial-steps in unit time (see [here][5]). 

[![enter image description here][2]][2]

What do we mean by "time"? In the language of LGV, we first fix positive integer $T$ and then set the weight of any path to be:

$$w(P)=\prod_{k=1}^{T}w(e_{k}),$$
where $P=(e_{1},e_{2},...,e_{T})$ and $e_{i}$ are diagonal edges as above. 

However, there are [applications][3] ([here][4]) where a version of LGV still "applies"(i.e. the LGV is used as an analogy) even though the paths are jumping in varying non-unit positive increments at each unit step time. In other words, a lattice path might jump two positive integers at time t: $P(t+1)-P(t)=2$ and three positive integers at some other time s: $P(s+1)-P(s)=3$. Here we still have a weight of a path
$$w(P)=\prod_{k=1}^{T-1}w(P(k),P(k+1)),$$
where $P=(P(1),P(2),...,P(T))$ and $P(i)$ is the position of the path at time t=i. For example, in the application [here][4] they take it to be an indicator
$$w(P(k),P(k+1))=1_{P(k+1)\geq P(k)}.$$




So it would be interesting to read of work done in LGV/Vicious-walkers and its generalizations that possibly include non-unit step. Of course, once one drops the unit-step requirement, one must also work with a more general definition of "intersection". 

I was thinking maybe with the bijection to [Young Tableaux][5], one can obtain a generalization in the Young Tableaux side even though there is no corresponding object at the Vicious walkers side.


  [1]: https://en.wikipedia.org/wiki/Lindstr%C3%B6m%E2%80%93Gessel%E2%80%93Viennot_lemma#:~:text=In%20mathematics%2C%20the%20Lindstr%C3%B6m%E2%80%93Gessel,of%20Lindstr%C3%B6m%20published%20in%201973.
  [2]: https://i.sstatic.net/Mw6zC.png
  [3]: https://arxiv.org/abs/math-ph/0608056
  [4]: https://arxiv.org/abs/cond-mat/0504417
  [5]: https://www.jstage.jst.go.jp/article/bjsiam/13/4/13_KJ00003574679/_article