Pick two integer sequences $d>a_n\geq a_{n-1}\geq\dots a_1\geq0$ and $d>b_n\geq b_{n-1}\geq\dots b_1\geq0$ where $d$ is an integer bound with following method:

Pick $a_n$ uniformly from $[0,d]$ and rest of $i\in\{1,\dots,n-1\}$ uniformly from $[0,a_{i+1}]$.

Do similar picking for $b_i$'s.

What is the distribution of $\max_i\log|a_i-b_i|$ and expected value of $\max_i\log|a_i-b_i|$ as a function of $d$ and $n$?