Payment channel networks like the Lightning Network (LN) boost cryptocurrency scalability but might have some privacy risks, as establishing a direct channel reveals user or entity relationships. This paper introduces a novel method using edge local differential privacy (LDP) applied during public channel creation to protect channel existence privacy. Our approach adds statistical uncertainty to the network graph, providing plausible deniability about direct links between specific users while ensuring channels remain usable for routing payments. We formally introduce edge LDP for this context, define relevant utility metrics focused on multi-hop payment feasibility, cost, and node centrality, and evaluate the privacy versus utility trade-offs using a snapshot of the Bitcoin LN.