diamond geezer

 Monday, September 09, 2019

Have you ever been for a random walk?

I decided to go for a random walk stating at Oxford Circus.
I used a coin, specifically a Mrs Tiggywinkle 50p, to decide which way to go.
At every junction it was Tiggywinkle for left, Queen's head for right.

n.b. This works fine at T-junctions and crossroads, but needs adapting for sideroads.
Sideroad on the left: Tiggywinkle for left, Queen's head for straight on.
Sideroad on the right: Tiggywinkle for straight on, Queen's head for right.
But my precise rules aren't important.
What is important is that, at every junction, which road to take was entirely out of my control.

I decided that my random walk would last exactly one hour.
Normally in one hour I can walk three miles.
I wondered whether my random walk would reach half a mile from Oxford Circus.

The mathematics of random walks is widely researched, and fascinating.
In two dimensions the general rule is that after N steps, you will be roughly √N steps from your starting point.
Central London's street pattern isn't regular enough for such a rule to apply exactly.
Also, pure random walking allows for going back the way you came, and I wasn't doing that.
But, anticipating one coin flip every minute, I reckoned half a mile ought to be unlikely.

If you imagine the local area divided up into quadrants, I could end up in any of them.
A lot hinged on the first few tosses.

I started by facing north up Regent Street.
This meant Tiggywinkle would take me left towards Marble Arch.
Instead I got the Queen, so headed right towards Tottenham Court Road.
On my second toss I got Tiggywinkle so avoided heading down Argyll Street into Soho.
On my third toss I got Tiggywinkle again and ended up in Fitzrovia.
And there I stayed.

My coin sent me twiddling round Market Place and Great Tichfield Street for a bit. It directed me east along Margaret Street, then pushed me further out towards Wells Street. It honed in on The Cartoon Museum. It shunted me off into a delivery zone round the back of the Sanderson Hotel. After ten minutes I was already quarter of a mile from my starting point, and making unexpectedly good progress. But could it last?

An important truth about random walks is that the path already travelled does not affect the path to come.
Just because I was heading firmly northeast didn't mean this would continue.
It was still impossible to predict where I'd be ten minutes later.
Essentially every toss of the coin starts a brand new random walk.

At twenty minutes I'd reached Riding House Street, with the Post Office Tower looming large. After thirty minutes I was walking down Tottenham Street for the third time, my coin having given me a looping sense of deja vu. After forty minutes I was back on Riding House Street again, but at the other end. I'd spent the last half hour walking repeatedly around a fairly compact slot of Fitzrovia - thankfully an interesting one, but it seemed all my lefts and rights kept cancelling out.

Eventually my coin led me back to Oxford Street and, against the odds, across it. At fifty minutes I was inside the Soho quadrant, somewhere round the back of the new Crossrail entrance, but the incursion didn't last. A couple of flips led me back into Fitzrovia, zigzagging towards and beyond Tottenham Court Road. I nearly made it to Bedford Square but as sixty minutes ticked round I had to make do with being in Morwell Street, a dreary backstreet. Unexpectedly I'd ended up precisely half a mile from Oxford Circus, but how much of a coincidence was this?

Obviously I tried again.
I got the bus back to Oxford Circus and prepared for another hour-long random walk.

Might this be the time to venture into Mayfair or Marylebone?
No, I started with two Queen's heads in a row and ended up in Soho.

After ten minutes I was in D'Arblay Street near the junction with Berwick Street. After twenty minutes I was back at the Crossrail entrance in Dean Street again. After thirty minutes I was back at the Crossrail entrance in Dean Street again. Random events do churn up some entirely improbable occurrences, not that any of these can be predicted in advance.

Perhaps the oddest thing is that I found myself walking clockwise around the whole of Soho Square, having thrown four Queens in a row... and then quarter of an hour later came back and did exactly the same thing again. Another peculiar thing is that after fifty minutes I was back in D'Arblay Street again, where I'd been forty minutes previously. A final unexpected thing is that my coin suddenly retraced my steps towards Oxford Circus so that I ended my walk at the top of Carnaby Street (where I'd been fifty-five minutes earlier). Even though I'd been half a mile away from Oxford Circus after 40 minutes, at the end of my hour I was almost back at the start.

According to the mathematics, you'll always get back to your starting point eventually, although it might take an infinite amount of time to get there.

If I put my two walks together on the same map, you can see how little of the surrounding neighbourhood I managed to cover in two hours.

I never once walked west of my starting point. Marylebone and Mayfair were never touched, nor the southern half of Soho. Several streets I visited three times, but most streets never once. The second walk often felt like it had got into a rut and was going round the same small area over and over, whereas the first walk was more varied and interesting. The first walk also led me along streets I'd never walked down before, past buildings and shops and plaques that helped make my random safari less of a pointless task. And although on each occasion I walked for three miles, I never got further than half a mile from where I started.

I couldn't have predicted any of this when I started, indeed there was absolutely nothing inevitable about where I ended up. A single coin toss flipping the other way would have changed the experience entirely. But a random walk is always an intriguing insight into how random processes operate, should you ever fancy giving it a try. I suspect it works best in city centres where the grid of streets is densest, but you could flip coins across a suburb, round a park or through a forest. Just don't get your hopes up for an exciting journey because willing the coin to send you in a particular direction never works. The Queen goes where she wants to go and Mrs Tiggywinkle has a mind of her own.

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