Simulation on random walk

Consider a Markov Chain as the random walk over the integers. Consider also Sn r.v. that takes values on S={-1,0,1} with probabilities beta,gamma and alpha. Start with X1=0. Then Xn=X_{n-1}+Sn, for n=2,3,... We also have that E(Sn)=alpha-beta, V(Sn)=alpha+beta-(alpha+beta)^2, thus E(Xn)=(n-1)(alpha-beta) and V(Xn)=(n-1)(alpha+beta-(alpha-beta)^2).

Here is my approach to implement this in R

#case 1: alpha>beta (I considered the particular case where alpha=1/2 and beta=1/4=gamma)

Nsim<-1000 

S<-c(0,sample(c(-1,0,1),Nsim-1,replace=TRUE,prob=c(1/4,1/4,1/2)))


Xn<-cumsum(S)


plot(Xn,pch=".",col="red")


abline(0,1/4,col="blue")

#case 2: alpha=beta (I considered the particular case where alpha=1/4=beta and gamma=1/2)

Nsim<-1000

S<-c(0,sample(c(-1,0,1),Nsim-1,replace=TRUE,prob=c(1/4,1/2,1/4)))


Xn<-cumsum(S)

plot(Xn,pch=".",col="magenta")

The case 1 works correctly, as you can see:

But case 2 does not and I don't know why see:

Why does this happen?

Also, the abline() in this case 2 should have slope 0, am I right?

Could someone please help?