Question about an ergodicity in economics paper
OP's Python code is computing one path in time and is not averaging over the paths. So it is not the value shown in the plot. In fact what the OP's Python code computes is $(1+\Delta x_H)^h(1-\Delta x_T)^t$ where $h$ denotes the number of heads and $t$ the number of tails tosses. It is closer to the red line which is the most likely path where $h=t$.
$$\mathbf E[x_{t+1}|x_t] = \big[(1+\Delta x_H)p_H+(1-\Delta x_T)p_L\big]x_t=(1+p_H\Delta x_H-p_L\Delta x_T)x_t=:ex_t,$$
and
$$\mathbf E[x_{t+1}|x_{t-1}]=\mathbf E[\mathbf E[x_{t+1}|x_t]|x_{t-1}]=e^2x_{t-1}.$$
So $\mathbf E[x_t|x_t]=e^tx_0$ exactly as shown in the paper plot. This can also be derived directly through the generating function of the expected $\frac{x_t}{x_0}$ which is, due to the independence of each toss, simply the product of all the dot product of payoff and its associated probability, i.e., $e^t$.