Simulating Ogata's Thinning Algorithm in R - simulation

Question

I am trying to implement Ogata's Thinning Algorithm exactly as given in Algorithm 3 in https://www.math.fsu.edu/~ychen/research/Thinning%20algorithm.pdf with the parameters they specify to generate Figure 2.
This is the code that replicates it verbatim.
# Simulation of a Univariate Hawkes Poisson with
# Exponential Kernel γ(u) = αe^(−βu), on [0, T].
# Based on: https://www.math.fsu.edu/~ychen/research/Thinning%20algorithm.pdf
library(tidyverse)
# Initialize parameters that remains the same for all realizations
mu <- 1.2
alpha <- 0.6
beta <- 0.8
T <- 10
# Initialize other variables that change through realizations
bigTau <- vector('numeric')
bigTau <- c(bigTau,0)
s <- 0
n <- 0
lambda_vec_accepted <- c(mu)
# Begin loop
while (s < T) {
# -------------------------------
# Compute lambda_bar
# -------------------------------
sum_over_big_Tau <- 0
for (i in c(1:length(bigTau))) {
sum_over_big_Tau <- sum_over_big_Tau + alpha*exp(-beta*(s-bigTau[i]))
}
lambda_bar <- mu + sum_over_big_Tau
u <- runif(1)
# so that w ∼ exponential(λ_bar)
w <- -log(u)/lambda_bar
# so that s is the next candidate point
s <- s+w
# Generate D ∼ uniform(0,1)
D <- runif(1)
# -------------------------------
# Compute lambda_s
# -------------------------------
sum_over_big_Tau <- 0
for (i in c(1:length(bigTau))) {
sum_over_big_Tau <- sum_over_big_Tau + alpha*exp(-beta*(s-bigTau[i]))
}
lambda_s <- mu + sum_over_big_Tau
# -------------------------------
# Accepting with prob. λ_s/λ_bar
# -------------------------------
if (D*lambda_bar <= lambda_s ) {
lambda_vec_accepted <- c(lambda_vec_accepted,lambda_s)
# updating the number of points accepted
n <- n+1
# naming it t_n
t_n <- s
# adding t_n to the ordered set bigTau
bigTau <- c(bigTau,t_n)
}
}
df<-data.frame(x=bigTau,y=lambda_vec_accepted)
ggplot(df) + geom_line(aes(x=bigTau,y=lambda_vec_accepted))
However, the plot I managed to get (running several times) for lambda vs time is something like this and nowhere near what they got in Figure 2 (exponentially decreasing).
I am not sure what is the mistake I am doing. It will be great if anyone can help. This is needed for my research. I am from biology and so please excuse if I am doing something silly. Thanks.