Jambalaya / Paella / Pulao

Some googling in relation to my previous post lead me to the Wikipedia page on Pulao/Pilaf, which they describe as :

a Middle Eastern and Central Asian dish in which a grain, such as rice or cracked wheat, is generally first browned in oil, and then cooked in a seasoned broth. Depending on the local cuisine it may also contain a variety of meat and vegetables.

Down at the bottom of the page under "See Also", they list Paella and Jambalaya. It hadn't struck me before how similar these three words are to each other. Surely they have a common root ? Wikipedia says that the root dish is Persian or Uzbek:

One of the earliest literary references to Pilau can be found in the histories of Alexander the Great when describing Sogdian (an Eastern Iranian province probably the birthplace of Alexander's wife Roxana and geographically situated in modern Uzbekistan) hospitality

It's easy to believe that it spread to Iberia to become Paella :

The Arabs probably introduced Pilaf to Iberia. Spanish paella was more than likely a standard Moorish method for cooking rice - with no wasted water, important in desert regions like North Africa.

And I'm convinced that Jambalaya is a distortion of the same word. But not everybody thinks so:

It may derive from the Spanish dish paella, possibly brought to Louisiana when Spain controlled the territory comprising the future Louisiana Purchase, although many other theories exist, including the notion that it is a combination of the words jambon (French for ham), à la (French for in the style of) and ya-ya (West African for rice).

Perhaps it's a mix of several words, if that's possible.

In related news, "some scholars think" that the words for Rice in many European languages is derived from the Tamil Arisi :

Indian scholars claimed that the word for rice in Western languages had a Dravidian root and that ris, riz, arroz, rice, oruza, and arrazz all came from arisi (Pankar and Gowda 1976).

Yes, discovering similar words in different languages is my idea of fun.