Suppose that I am placing a market order directly in the order book of an exchange. For market orders, it seems quite clear that larger orders obtain larger spreads due to the fact that - without loss of generality - a large market buy order must reach further into the limit sell orders of the order book than would a smaller market order.
Assuming that this initial premise is true (and if it's not, please tell me!), what would be the nature of this relationship? Would it be linear? Quadratic? Logarithmic? Any insights would be appreciated, including links to existing research on the subject.