I'm afraid I don't have access to neat statistical mapping of the precise correlation between the number of slaveholding families in a given area, the average size of slaveholdings, and crime rates. It may well be that someone has done some statistical research in this area and has a more numerically precise model than I can offer you.

But the answer is yes. In the antebellum United States, slaveholding states had significantly more violence within them than free states (which is not to say that the North did not have its own incidents of appalling violence).

Your question, however, touches on an ambiguity here with the conflation of "violence" and "violent crime." Slavery was, at its core, a violent institution. But the law in slaveholding states (and earlier North American colonies) only recognized exceptional cases of violence against enslaved people by their owners as criminal (and even then it did so exceedingly rarely). For instance, in Colonial Virginia, a slaveholder who killed an enslaved worker needed only to explain that he did so in the process of disciplining them to be spared from any charges. No southern states recognized rape as a crime if the victim was an enslaved woman. This means that the arrival of, say, the younger son of a very rich South Carolina slaveholding family moving to Deer Creek, Mississippi with 150 enslaved workers could lead to a huge increase in violence without a corresponding increase in crime. It looks like you're more curious about, if you'll pardon the phrase, "white-on-white crime" in the antebellum South, and slavery's relation to it, but the fact that slave labor depended on violence for its daily reproduction is a key bit of context.

As far as I know, there aren't statistical estimates of rates of domestic abuse in the antebellum south, especially given that their definitions were different from our own, but there is plenty of suggestive evidence. There's a great deal of testimony speaking to frequent domestic abuse in white slaveholding families, with some sources suggesting familiar cycles of violence and misery. One antebellum account of the Bird family, whose patriarch was a respected physician, described how he would mercilessly beat his wife, Eliza, in the streets. The observer noted that the couple's daughter had "the united temper of both parents.... [and was] most wretchedly managed, sometimes whipped severely when her mother is in a passion, and then indulged in every whim as if to make up for the severity that had been used."

Sarah Gayle, wife of the governor of Alabama in the 1850s, wrote of an acquaintance whose husband's abuse was so brutal and terrifying that it sent her fleeing her home into a nearby swamp.

Lewis Wallace, who had been enslaved in Mississippi and recounted his experiences to an interviewer with the WPA, remembered his former owner as an abusive alcoholic, who would "grab old Missus by de hair of her head an' drag her up and down de long front gallery." I could fill many pages with similar anecdotes.

Historian Nell Irvin Painter has argued that cases of sexual abuse and incest within white slaveholding families were much more frequent than our archival materials suggest. More on why she believes this in a moment.

Here's one revealing statistic: in Florida in 1830, the murder rate among white residents was roughly 70 per 100,000. In New England that number was roughly 1.5 in 100,000.

White southern men of all social classes murdered each other over seemingly trivial conflicts with appalling frequency, which plays a large role in that dramatic statistic above. When future Senator James Henry Hammond attended South Carolina College in the 1820s, two of his classmates killed each other in a duel. What prompted this tragic, bloody end? The two young men had both reached for the same tray of trout at the college's mess hall at the same time, and neither one was willing to give up precedence to the other. When P.T. Barnum's travelling exhibition came to Charleston, South Carolina in the 1850s, two local gentleman nearly dueled each other over the question of whether the Fiji Mermaid was real. Less distinguished men often murdered each other in bar brawls, or at least gauged each other's eyes out if the fights were not fatal. If you wanted to provoke a duel to the death in the antebellum South, you could easily do so by tweaking another man's nose.

What is slavery's role in all of this?

Nell Irvin Painter argues in a famous essay, Soul Murder and Slavery, that "the values and practices of slavery, particularly the use of violence to secure obedience and deference, prevailed in white families as well." There was no way to compel slave labor without the constant use of violence, and the habits of mind and action engendered by such a necessity could not simply be changed into and out of like a costume. This, in her view, caused severe psychological damage to white southerners, even as they actively and enthusiastically tortured and traumatized millions of enslaved workers.

Kenneth Greenberg, a historian of "honor," argues that among white southern men, trivial challenges to status and honor had life-or-death stakes because in Southern political culture there was no in-between status joining "free" and "enslaved." This was also a byproduct of the South attempting to be a republican slave society, which required a class of citizens who shared some ostensible equality in the political realm. Free white men had honor and enjoyed political equality with each other in part because of that shared honor. Enslaved people did not. Honor, he argues, was also a profoundly superficial lens through which to view the world. Honor was entirely about appearances, rejecting the idea that appearance might mask a deeper truth. We today might be said to live in a "dignity culture," where a great deal of our self-worth and esteem comes from within, and is ostensibly immune to the opinions of others. Honor is not like that. Honor is about what other people think of you. It's about them accepting the image of yourself that you put forward into the world (remember: your nose is the first part of you that enters a room!). Expressing contempt for that image -- by calling someone a liar, by tweaking their nose, by saying they are are wrong about the authenticity of the Fiji Mermaid -- was an action with high high stakes in a slave society.

I think Greenberg and Painter are arguing very similar points: that the violence so central to slavery rippled dramatically through white southern culture. Painter is a little more broad in her hypotheses where Greenberg focuses on a more specific concept and its expression. But I largely agree with both. Slavery was the central material reality of most white southerner's lives, and it made white people more violent overall because it depended on violence to function.

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Sources:

Nell Irvin Painter, Soul Murder and Slavery. Published in Southern History Across the Color Line

Thavolia Glymph, Out of the House of Bondage: The Transformation of the Plantation Household

Edward Baptist, The Half has Never Been Told

Edward Ayers, Vengeance and Justice: Crime and Punishment in the 19th Century American South

Kenneth Greenberg, Honor and Slavery: Lies, Duels, Noses, Masks, Dressing as a Woman, Gifts, Strangers, Humanitarianism, Death, Slave Rebellions, the Proslavery Argument, Baseball, Hunting, and Gambling in the Old South (surely one of the great history book titles of all time)