Leverage increase returns, but also risks, ie, the least you can pay, the greater the opportunity to profit, but also the greater the chance you will be underwater.
Leverage is given by the value of your asset (the house) over the equity you put down. So, for example, if the house is worth 100k and you put down 20k, then the leverage is 5 (another way to look at it is to see that the leverage is the inverse of the margin - or percentage down payment - so 1/0.20 = 5).
The return on your investment will be magnified by the amount of your leverage. Suppose the value of your house goes up by 10%. Had you paid your house in full, your return would be 10%, or 10k/100k. However, if you had borrowed 80 dollars and your leverage was 5, as above, a 10% increase in the value of your house means you made a profit of 10k on a 20k investment, a return of 50%, or 10k/20k*100. As I said, your return was magnified by the amount of your leverage, that is, 10% return on the asset times your leverage of 5 = 50%. This is because all the profit of the house price appreciation goes to you, as the value of your debt does not depend on the value of the house. What you borrowed from the bank remains the same, regardless of whether the price of the house changed.
The problem is that the amplification mechanism also works in reverse. If the price of the house falls by 10%, it means now you only have 10k equity. If the price falls enough your equity is wiped out and you are underwater, giving you an incentive to default on your loan.
In summary, borrowing tends to be a really good deal: heads you win, tails the bank loses (or as happened in the US, the taxpayer loses).