E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
To meet one in the Dukes Walk. An invitation to fight a duel. In the vicinity of Holyrood House is a place called the Dukes Walk, from being the favourite promenade of the Duke of York, afterwards James II., during his residence in Scotland. This walk was the common rendezvous for settling affairs of honour, as the site of the British Museum was in England.
If a gentleman shall ask me the same question, I shall regard the incivility as equivalent to an invitation to meet him in the Dukes Walk.Scott: Bride of Lammermoor, chap. xxxiv.