(TMZ) -- Kim Kardashian's robbers have strong ties to the illegal diamond trade in Belgium and now the country's federal police force is on what is probably a futile hunt to track the bling down.

Sources with knowledge of Belgium's "black market" for diamonds tell us several men arrested for the Paris robbery have ties to the market. There's a diamond district in Antwerp and shady jewelry shops do business just outside the perimeter ... that's likely where Kim's jewelry ended up.

Law enforcement sources in Belgium tell TMZ their federal police force is investigating, but TMZ's sources say ... since more than 3 months have passed it'll be next to impossible to get the bling back. Stuff on the illegal market just moves around too quickly.

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