Complete local expansion of the availability function in random sequential adsorption of aligned squares at low...

We consider random sequential adsorption (RSA) of aligned squares and derive the low-coverage expansion of the availability function α(q), the fraction of positions accessible to an additional square, up to fourth order in the coverage q. At low coverage, the reduction of available space can be understood in terms of geometric overlap between exclusion regions created by previously deposited squares. A single square blocks a finite area; pairs of squares may have overlapping exclusion zones, reducing the total blocked area; similarly, three and four squares can share a common overlap region, leading to higher-order corrections. These contributions can be systematically accounted for through an inclusion-exclusion expansion based on the geometry of overlapping exclusion regions, with alternating signs dictated by inclusion-exclusion. The expansion terminates exactly at fourth order, since no more than four deposited squares can simultaneously overlap the exclusion region of a trial insertion. The coefficients are obtained by explicit enumeration of all such geometrically admissible configurations and are further confirmed by numerical simulations on a discrete lattice, showing agreement within statistical uncertainty.