In a Hash Array Mapped Trie (HAMT), the process of finding a key associated with a value involves the following steps:

Hash the key: The key is hashed to produce a hash value, which is used to index into the array part of the structure.

Look up the array: The array is used to determine which node in the trie represents the portion of the key corresponding to the hash value.

Traverse the trie: The trie is then traversed starting from the node found in step 2, using the remaining portion of the key, until the leaf node corresponding to the complete key is found.

Retrieve the value: The value associated with the complete key is stored at the leaf node, and can be retrieved once the leaf node is found.

The hash function used to hash the key and the implementation of the trie structure determine the efficiency of the look up process. The use of a hash table makes the look up process faster compared to other forms of tries, as it allows for constant-time access to the first node in the trie corresponding to the portion of the key.
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