What are proximity searches?

As a search operator
In the main search bar, you can write an exact query in double quotes with the search operator tilde (~) with a number, at the end of your query. It corresponds to the maximum number of operations (insertions, deletions, substitutions and transpositions) on terms needed to make one phrase match the other.
Examples:
“the cat is blue” -> “the small cat is blue” (1 insertion = fuzziness is 1)
“the cat is blue” -> “the small is cat blue” (1 insertion + 2 transpositions = fuzziness is 3)
"While a phrase query (eg "john smith") expects all of the terms in exactly the same order, a proximity query allows the specified words to be further apart or in a different order. A proximity search allows us to specify a maximum edit distance of words in a phrase." (source: Elastic).
Example: "fox quick"~5 (this query will catch "quick brown fox", "quick brown car thin fox" or even "quick brown car thin blue tree fox"
The closer the text in a field is to the original order specified in the query string, the more relevant that document is considered to be. When compared to the above example query, the phrase "quick fox" would be considered more relevant than "quick brown fox"(source: Elastic).
In Batch Searches
When you turn on 'Do phrase matches', you can set, in 'Proximity searches', the maximum number of operations (insertions, deletions, substitutions and transpositions) on terms needed to make one phrase match the other.
“the cat is blue” -> “the small cat is blue” (1 insertion = fuzziness is 1)
“the cat is blue” -> “the small is cat blue” (1 insertion + 2 transpositions = fuzziness is 3)
Example: "fox quick"~5 (this query will catch "quick brown fox", "quick brown car thin fox" or even "quick brown car thin blue tree fox"

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What is fuzziness?

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As a search operator

In Batch Searches