walkers.R

# measure the average number of steps a random walker takes in two
# dimensions to return to origin for the first time by simulating
# the random walk multiple times.
library(tidyverse)

# walk through an integer grid one step at a time
# report the number of steps on first return to the origin
steps_on_first_return <- function(max_steps = 10000L) {
  x <- 0L
  y <- 0L
  # directions 1 through 4 are right, left, up, and down.
  dx <- c(1L, -1L, 0L, 0L)
  dy <- c(0L, 0L, 1L, -1L)

  for (step in seq_len(max_steps)) {
    direction <- sample.int(4L, size = 1L)

    # go 1 unit in 1 direction only
    x <- x + dx[direction]
    y <- y + dy[direction]

    # if returned to orgin, report steps
    if (x == 0L && y == 0L) {
      return(step)
    }
  }

  # the walker has not returned within the step limit;
  # its return steps is unknown
  NA_integer_
}

# as.integer(Sys.Date()) for today gives me the seed
set.seed(20706)
n_walks <- 5000L
max_steps <- 10000L

walks <- tibble(walk = seq_len(n_walks)) |>
  mutate(
    return_steps = map_int(walk, \(i) steps_on_first_return(max_steps)),
    returned = !is.na(return_steps)
  )

return_summary <- walks |>
  summarise(
    n_walks = n(),
    step_limit = max_steps,
    n_returned = sum(returned),
    n_unfinished = sum(!returned),
    proportion_returned = mean(returned),
    mean_steps_all = mean(return_steps),
    mean_steps_among_returns = if (any(returned)) {
      mean(return_steps[returned])
    } else {
      NA_real_
    }
  )

return_summary

# show first-return times only for walks that returned within the step limit.
walks |>
  filter(returned) |>
  ggplot(aes(x = return_steps)) +
  geom_histogram(bins = 10, fill = "steelblue", color = "white") +
  labs(
    title = "Steps to first return",
    subtitle = paste("Walks returning within", max_steps, "steps"),
    x = "Steps",
    y = "Number of walks"
  ) +
  scale_x_log10() +
  theme_minimal()