Generating a random walk that can move right, left, up and down

Question

Suppose that we have a dice with 4 faces. If the dice lands on:

1. the person will move to the right
2. the person will move to the left
3. the person will move upward
4. the person will move downward

The simulation has n steps and the position of the person updates on every step. How can I generate this random simulation?

test[n_] := Accumulate[RandomChoice[{{-1, 0}, {1, 0}, {0, 1}, {0, -1}}, n]]

How can I show the first/last point on the simulation if I use a function similar to ListLinePlot[test[10]]?

Answer 1

You might implement your random walk something like this.

walk[n_] := 
  Prepend[
    Accumulate[RandomChoice[{{-1, 0}, {1, 0}, {0, 1}, {0, -1}}, n]], 
    {0, 0}]

trial[n_] := 
  Module[{path, start, end},
    path = walk[n];
    start = First @ path;
    end = Last @ path;
    ListLinePlot[path,
      Epilog -> {PointSize[Large], Point[start], Red, Point[end]},
      AspectRatio -> Automatic]]

SeedRandom[3]; trial[25]

Answer 2

Clear["Global`*"]

SeedRandom[123];
walk[n_Integer, init_ : {0, 0}] := Module[{
   up = {0, 1},
   dn = {0, -1},
   right = {1, 0},
   left = {-1, 0}
   },
  steps = RandomChoice[{left, right, up, dn}, n];
  FoldList[Plus, init, steps]
  ]

walk[10, {1, 1}]  (*starting at {1,1} with 10 steps *)

{{1, 1}, {2, 1}, {2, 0}, {3, 0}, {2, 0}, {2, 1}, {2, 2}, {2, 3}, {2,
4}, {1, 4}, {0, 4}}

---

Visualization:

With[{n = 30},
 path = walk[n, {2, 2}] // Echo;
 ListLinePlot[Partition[path, 2, 1], AspectRatio -> Automatic
   , PlotStyle -> (ColorData["Rainbow"][#] & /@ Subdivide[n - 1])
   , PlotLegends -> Placed[BarLegend[{"Rainbow", {0, n}}], After]
   , PlotRangePadding -> Scaled[.15]
   , AspectRatio -> Automatic
   , Epilog -> {AbsolutePointSize@Large, Black
     , Point@path[[1]], Circle[path[[1]], 0.2]
     , Red, Point@path[[-1]], Circle[path[[-1]], 0.2]
     }
   ] /. Line[a__] :> Arrow[a, 0.05]
 ]

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