A judge will decide Tuesday whether an Illinois man convicted of killing his wife and three children deserves a new trial, based in part on claims that the behavior of lawyers next door during Drew Peterson's murder trial made it impossible for the man to get a fair trial.Christopher Vaughn was convicted of fatally shooting his family in their SUV so he could start a new life living in the Canadian wilderness. His September trial overlapped with the high-profile trial of Peterson, the former suburban Chicago police officer convicted of murdering his third wife.