"Show me your papers": The statistical version
Several commentators have claimed the Arizona's new immigration law to be unconstitutional. Rightly so. The most controversial provisions are vague in specifying when "reasonable suspicion" exists. It is clear that the only way for a police officer to suspect somebody is their race, which would violate the Equal Protection clause (14th amendment). The law also violates the fourth amendment right against unlawful seizure.
However, I do think there is a way to ask someone to show their paper without violating the constitution. State law enforcement should choose people on the streets at "random" and ask them for paper. I am not using the word random to mean arbitrary. I am instead using random to mean the mathematical definition of random. The government decides to ask say thousand folks for their papers every day and the candidates for the check are decided by some random number generator --- something that says check the Nth person who drives through the intersection of X and Y cross street. The candidates, or say victims, would be uniformly distributed across the state, thereby not violating the 14th amendment.
The tricky issue is the 4th amendment. How can random searches pass the "reasonable suspicion" test? Well, we can rely on probability and the definition of reasonable. Everyone knows that there are undocumented immigrants in Arizona. They constitute around 5% of Arizona's population. The probability that any randomly chosen person is an undocumented immigrant is therefore 5%. While 5% might be a low number, think about it. If it were 30%, would it satisfy the "reasonable suspicion" requirement? Who gets to decide at what point pure random checks become constitutional? Lawmakers or a judge?