  In this manuscript, we delve into the exploration of the first and second
Zagreb connection indices of both polyomino chains and random polyomino chains.
Our methodology relies on the utilization of Markov chain theory. Within this
framework, the article thoroughly examines precise formulas and investigates
extreme values. Leveraging the derived formulas, we further explore and
elucidate the long-term behavior exhibited by random polyomino chains.
