Beyond urban boundaries: Percolation reveals land-consumptive growth in India

Cities often extend beyond their administrative boundaries, complicating assessments of urban sustainability. Using Global Human Settlement Layer data for India's 50 largest urban agglomerations (1975--2025), we develop a percolation-based framework for urban delineation. Within district-based regions, we apply unrestricted DBSCAN to spatial coordinates of 100-m cells with positive population and built-up surface. We treat neighborhood distance as the percolation control parameter and the proportion of eligible cells in the largest connected cluster as the empirical order parameter. Sharp increases in this proportion reveal characteristic percolation transitions. Post-transition connection lengths declined (median:650m to 350m), indicating that previously disconnected settlements coalesce at progressively shorter spatial distances alongside increasing urban land consumption. Urban footprint increased from 447.6 to 1,382.7km$^2$, while population densities declined. SDG11.3.1 land-consumption-to-population-growth ratios generally exceeded unity. Zipf and allometric relationships describe cross-sectional urban structure, but land-consumptive growth shows little relationship with city size. Urban delineation thus shapes sustainability assessments.