Could you please help to simulate in R bivariate Normal distribution (B(x), B(y)) with mean zero and covariance given by cov(B(x),B(y))=min (x,y)

My main challenge is in specifying the covariance min(x,y) in R. I come up with the below code but obviously the code is not correct

library(mvtnorm)
# Set parameters
n <- 100  # Number of time periods
x <- seq(0, 1, length.out = n)  # Values of x at each time period
y <- seq(0, 1, length.out = n)  # Values of y at each time period
cov_matrix <- matrix(0, n, n)  # Initialize the covariance matrix

# Compute covariance matrix
for (i in 1:n) {
  for (j in 1:n) {
    cov_matrix[i, j] <- min(x[i], y[j])
  }
}

# Generate samples from multivariate normal distribution
samples <- mvtnorm::rmvnorm(500, mean = rep(0, n), sigma = cov_matrix)

# # Split samples into B(x) and B(y)
 b_x <- samples[, 1:(n/2)]  # B(x) values
 b_y <- samples[, (n/2+1):n]  # B(y) values
 
 # Plot Bivariate Brownian Motion
 plot(x = b_x, y = b_y, type = "l", xlim = c(-0.5, 0.5), ylim = c(-0.5, 0.5), 
      xlab = "B(x)", ylab = "B(y)", main = "Bivariate Brownian Motion")