CANNES, France — Jim Jarmusch and Bill Murray’s long-running relationship began, fittingly, with a cup of coffee.

In the early 1990s, Jarmusch was walking north on Manhattan’s Columbus Avenue (“not my neighborhood,” he notes) when he noticed a guy walking toward him. That’s Bill bleeping Murray, Jarmusch said to himself.

“Bill walked right up to me and said, ‘You’re Jim, right?’” Jarmusch recalls. “And I said, ‘Yeah. You’re Bill Murray.’” And then he said, “You want to get a cup of coffee?”

They popped into a diner and after chatting for half an hour, Murray announced, “I gotta go. Nice talking to you.” Jarmusch was gobsmacked by the random meeting.

“We didn’t talk again for years but I told my friends: I met Bill Murray,” he says.

Murray, who has had enough chance encounters in his life to last a thousand Groundhog Days, grants some “tingling” in his brain at Jarmusch’s memory. But Murray long ago gave up remembering how he knows who he knows.

“I don’t recollect much,” Murray says. “When somebody asks me ‘How’d you meet?’ I say I really don’t know.”

That Murray and Jarmusch would find each other was probably fated. Both have made deadpan a high art form, finding sublimity in the bone-dry. You wouldn’t want to play either of them in poker.

They’ve made three films together, starting with the 2003 black-and-white vignette anthology “Coffee and Cigarettes.” Murray played a waiter whose two customers, the Wu-Tang Clan’s RZA and GZA, enthusiastically recognize him. Then came the 2005 drama “Broken Flowers,” a high point for both, in which Murray played, in Julie Delpy’s words, “an over-the-hill Don Juan” whose idleness is shattered by the news that he fathered a son 20 years ago.

Now, in “The Dead Don’t Die,” Jarmusch’s wry but impassioned zombie tale, Murray plays the veteran sheriff of a small town called Centerville populated by Jarmusch regulars, among them Tilda Swinton, Adam Driver, Chloe Sevigny, Tom Waits and Steve Buscemi. Jarmusch said he was moved to write something like “Coffee and Cigarettes” with “a kind of ridiculousness to it.” But it’s also a zombie parable about issues of urgent seriousness to Jarmusch: digital-age distraction and climate change.

Shortly after “The Dead Don’t Die” opened last month’s Cannes Film Festival, Jarmusch and Murray sat down to discuss their collaborations together. They drank coffee.

For “The Dead Don’t Die,” Jim wrote a part specifically for you. What was your first impression?

Murray: I didn’t know which part was mine. I kept thinking: I hope it’s this police chief guy because I thought it was so funny. Then when I started doing the movie, I realized, “Holy God, I missed this completely. Adam’s stuff is the funny stuff.” I realized I had it all wrong and I had to play it straight here. And I love playing straight. It’s the most serving thing you can do. You just lob them out there and don’t cash in your chips.

The premise of “The Dead Don’t Die,” in which the Earth is irrepealably damaged by “polar fracking” doesn’t seem so far-fetched. Do you each think much about the future of the planet?

Jarmusch: All of this other stuff is basically meaningless. All these politics, all this stuff doesn’t mean anything when in 12 years, you’ll be trying to get water for your child to drink. It’s very bad. Everyone’s just oblivious. They’re distracted, I guess. That makes me very sad. We have consciousness, we’re here. We have beautiful things to still appreciate but for how much longer I’m not sure.

Murray: To me, the apocalypse is if the human fabric just continues to shred itself by this divisiveness, this opposition mentality that’s developed. It’s basically maintaining manners as much as you can, politeness with people. If people are arguing, I get just myself in the middle of it as a complete clown, just to stop it. I try to be this neutralizing force that jumps in and takes all the focus. ‘Cause I can take it. I can dish it out, too. But I can take it.

Imagining the worst doesn’t help me. I’ve found that for me, it’s an energy I can put someplace else. Like the man said, it’s in the details, taking care of the little things. I started picking up trash on the street. If it’s right next to me, I pick it up and put it in the trash. Some sort of maintenance. It’s a little thing but it makes me feel like a citizen.

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