Around 7.45pm on Friday, 2 June, 2023, officers were called to a report of a disturbance on Leith Street near to the junction with Greenside Row.
Peter Mullen, 30, of Edinburgh, suffered serious injuries and was taken by ambulance to Royal Infirmary of Edinburgh, where he died a short time later.
Following a post mortem examination, his death is being treated as murder.
His relatives are aware and a report will be sent to the Procurator Fiscal.
Enquiries have so far revealed a number of people were involved in the disturbance and a 41-year-old man was also taken to hospital for treatment.
A 46-year-old woman has been arrested and charged in connection with a minor assault.
She is expected to appear before Edinburgh Sheriff Court at a later date and a report will be sent to the Procurator Fiscal.
Detective Chief Inspector John Morrison said: “Our thoughts are very much with Peter’s family and friends at this very difficult time.
“We have a dedicated team of officers working on this investigation and I would like to reassure the local community that everything is being done to trace the person responsible.
“Extensive enquiries are ongoing and officers are continuing to gather CCTV footage from the surrounding area.
“We are aware that the area was busy at the time of the incident and we are again encouraging anyone who may have seen or heard a disturbance, to please get in touch.
“We have set up a website that gives members of the public access to a form to send information directly to us.
“Anyone with any CCTV, dashcam or any other footage that they think would assist the enquiry can also send this via this link.
“You can select to remain anonymous when submitting information.
“I’d like to thank the public for their assistance so far and ask anyone with information who hasn’t already spoken with officers to please call 101 quoting incident number 4094 of 2 June, 2023.
“Alternatively, please call Crimestoppers on 0800 555 111 where information can be given completely anonymously.”
The police have set up a link to a Major Incident Public Portal (MIPP) that allows members of the public to send information directly.