When someone is in a foreign country and is unfamiliar with the people, the culture, and the system of justice of the nation they are visiting, it is easy to feel that you are safely beyond the reach of those in the host country, that no one knows you are there, that you can come and go as you please, and that no one can find you.
Part of the purpose of assimilating immigrants to a nation they are joining, is to remove this sense of anonymity, of being an invisible roamer, and to instill the sense that each citizen is expected to contribute to the cultural and societal structure of the country and that if they break the rules, that they will be pursued and caught and punished.
You can expect a very high crime rate from illegal slipping across the border. Not only do they embrace criminal behavior in the simple act of coming across the border illegally, they feel they can do whatever they want in this country and slip back across the border to hide, if they so choose, or simply stay here since they are here illegally and can't be tracked down.
There is another reason you can expect a high crime rate from illegals. Many have been convicted of crimes in their country; which makes it impossible for them to get documents to come into the US legally.