In my case.. yes. I just joined current company since Aug 1. HR called my references on 2-3rd week lol. Luckily I had a great relationship with my previous boss, which he is more like a mentor to me right now.

I am joining a direct competitor and my previous boss still say good words about me, he just told me when I catchup with him the other week. But I think it is okay, as I switch my career path from marketing > partnership.

I joined current company on 2nd of August. I left my previous company (2 year and 3 months) and joined current company. It's a direct competitor actually.
Hence, my current HR did not do contact my references before that. 2-3 weeks ago, current HR contacted my previous HOD which I put at my references list. Ask me about my attitude, performances, would he hire me back those question etc.

Luckily my previous HOD has a good relationship with me, which right now more like a mentor or A.K.A father to me. He understood my situation why I choose to leave and he still managed to tell all good stuff to my current employer. He just told me that my current HR contacted him when I went to catch-up dinner with him at the last 2 week. Current HR told my ex boss that it was a standard procedure to contact references if the new employee joined from competitor/similiar indusrty.