I disagree, the number of moves necessary to live in the center is not that much higher than in the corner. Assuming sequential detective moves, meaning the fugitive needs two eyes, it takes …
6 moves in the corner
8 moves on the side
10 moves in the center
Hence I believe the detectives would be ill-advised if they ignore the center for too long. And likewise, larger board sizes mean more space for the fugitive to hide / more space for the detectives to cover.