SHREVEPORT, La. — In a much-discussed dissent from the Supreme Court’s ruling on lethal injection last week, Justice Stephen G. Breyer laid out the problems, as he saw them, with the death penalty. Among them was “arbitrariness in application,” including how simple geography can determine whether someone convicted of murder would be sentenced to death.

“Between 2004 and 2009,” Justice Breyer wrote, “just 29 counties (fewer than 1 percent of counties in the country) accounted for approximately half of all death sentences imposed nationwide.”

Caddo Parish, here in the northwestern corner of the state, is one of these counties. Within Louisiana, where capital punishment has declined steeply, Caddo has become an outlier, accounting for fewer than 5 percent of the state’s death sentences in the early 1980s but nearly half over the past five years. Even on a national level Caddo stands apart. From 2010 to 2014, more people were sentenced to death per capita here than in any other county in the United States, among counties with four or more death sentences in that time period.

Robert J. Smith, a law professor at the University of North Carolina whose work was cited in Justice Breyer’s dissent, said Caddo illustrated the geographic disparity of capital punishment. But he said this analysis did not go far enough. Caddo, he noted, has bucked the national trend in large part because of one man: Dale Cox.