A random walk in two dimensions will eventually take you where you want. It may take a very long time, depending on the number of choices, and the distance to cover, and luck. But you will always, sooner or later, emerge from a maze, provided you haven’t died of cold or thirst in the meantime. In two dimensions, all roads lead to Rome.
This tunnel was even narrower than the last one. It went on for ever.
A random walk in three dimensions is different. In a three-dimensional maze, the probability that you’ll end up where you want is about — Petrie tried to remember the exact figure — 0.35, that’s it, one in three.
I’m going to die here, right inside one of my knots.
McCrea and Whipple, that was it. Proceedings of the Royal Society of Edinburgh, 1940. The Nazis were about to invade Britain and these cool dudes were calculating random walks, like playing bowls with the Armada on its way.
After ten minutes he thought to give up, crawl — backwards! — back along its length. Something stupidly jumped into his head:
It may be proper likewise to mention to the benighted traveller, that when he falls in with bogles, whatever danger may be in his going forward, there is much more hazard in turning back.
R. Burns, footnote to Tam O’Shanter, 1710
The tunnel curved slightly to the right, giving him a ten-metre line of sight. He would crawl that distance and then give up.
Ten metres on, the tunnel seemed to be opening out. He gave it another ten, and then twenty, and then the torchlight was showing a large chamber strewn with boulders. Reaching it, he found it was about nine feet high and adorned with thousands of little needle-like stalactites and stalagmites. He stood up and stretched.
Leading off from this chamber was a curving man-sized tunnel. If he wasn’t lost, and if Svetlana’s map was right, this would lead to Tyson’s Autobahn.
It did, and Petrie almost wept with relief. He shouted, ‘Yes!’ and, ‘Yes! Yes! Yes!’ echoed back at him, fading into silence.
The Autobahn, an ancient bedding plane uplifted by whatever forces had created the Tatras, was a lot steeper than he had expected it to be. It was about twenty metres broad and ten high, patterned with stony icicles on floor and roof, many of them as thick as tree trunks. It stretched beyond the range of Petrie’s lamp.
Again he momentarily switched off; again, pitch black. Freya! Where are you?
A kilometre, Svetlana had said. He set off smartly, his breath echoing in the long chamber. Gradually, as he climbed, the roof became lower. He thought he glimpsed …
Voices!
Petrie froze, switched off his helmet lamp and stood in the pitch black. Not pitch black. Lights, at the foot of the Autobahn. Half a kilometre away, but it was hard to judge distances.
A candlelit procession. Skiers gliding down a piste, torches lighting the snow. A Viking funeral.
None of these. Men with guns, hunting him.
He switched on his lamp, turned to run.
Men shouting, the tunnel efficiently transmitting the voices so that they seemed only yards away.
Bullets would follow. Petrie began to swerve, keeping big stalagmites between him and his pursuers when he could. The ground levelled and sloped down; another fifty metres and he would be out of sight. A smattering of gunshot and little chips of limestone buzzing off stone icicles. One of them hit him hard on the cheek, drew blood. A last swerve and he was running downhill and there it was, seven metres down a shaft, the sump, a turquoise pool of infinite depth with a red nylon rope descending into it, attached to the vertical rock by pitons, and demons waiting for him under the smooth surface.
Petrie stared, horrified, at the water, still as death. Nothing was going to induce him into that pool.
Ah, Tam! Ah, Tam! thou’ll
get thy fairin! In hell,
they’ll roast thee like a
herrin.
A shouted command. They were much closer. Or was it the acoustics? Petrie scrambled down, gripping the rope, slipped, cried with pain as the friction burned the palms of his hands. Then his ankles were into the icy water, and his waist and his chest. He took a deep breath and then his head was under and he was hauling himself along clumsily, upside down, water getting into his nose.
He made no attempt to time the underwater journey. He didn’t try to estimate how far the sump went down. But it levelled at some depth and he was pulling himself along frantically, his helmet bumping against the rock. He wondered if Freya had gone much further along the sump, whether he would bump into her drowned body. Stars began to explode in his eyes. Random walks, knots, soldiers, Freya, Hapsburg castles, alien signals, all went from his mind and were replaced by a single, burning focus: the red nylon line, winding through branching tunnels. And then the rope was curving upwards and there was light and he broke the surface, whooping and gasping.
Freya was perched on a boulder, looking like a sodden elf. Her helmet torch dazzled him.
‘You took your time,’ she said.
He heaved himself on to a ledge, flopped out on it for a few seconds. Then he sat up, still gasping. ‘They’re on the Autobahn. They shot at me.’
Freya scrambled around, picked up a fist-sized rock and started to hammer at a piton. Petrie, weak at the knees, scrabbled around for another stone.
‘Tom, someone’s on it.’
The rope had gone taut, as if a fish had been hooked on it.
Petrie said, ‘Jesus.’
But now the piton was loose and in a minute they had the grim satisfaction of seeing the guide rope spiral down towards the bottom of the sump. Petrie felt a surge of guilt at his own hope that the soldiers using it would drown.
And now they were in a cavern with limestone stalagmites the size of tree trunks and branches leading off like exits from Piccadilly Underground.
‘I know the way,’ Freya said. ‘I’ve been along it.’ She led Petrie, streaming water, along a high tunnel, through a knee-deep stream. Ahead, faint light was scattering off the tunnel wall. They switched off their lamps. Human chatter began to echo. And then there was a short, dry shaft, brilliantly lit from the other end. Freya put a finger to her mouth and crawled along it. Petrie followed, and found himself looking down into a vast natural cathedral with flowstones, fountains and millions of stone icicles, frozen in brown, white and orange limestones. The cathedral was glowing from the spotlights scattered around its walls. About twenty people were clustered round a guide.
‘This is the second party I’ve watched,’ Freya whispered. ‘I think the woman at the left is there to pick up stragglers.’
‘Wait until they’ve moved off, then catch up, with apologies. If anyone asks about our clothes, we’ve been scuba diving.’
But Freya, scanning the cavern, seemed not to have heard.
‘Freya, we can’t hang about. There could be strong swimmers amongst the soldiers. We may only have a couple of minutes’ start.’ He looked back fearfully at the black hole of the sump.
‘I hope they all drown,’ she hissed. Then she cuddled into him like a little wet kitten and whispered, ‘Isn’t this beautiful?’ and Petrie tried to suppress a fit of giggling hysteria.
35
Death Squad
Gibson shambled reluctantly into the refectory. The entire team was present and the table was laid with toast, hard-boiled eggs, butter and jam. The croissants looked and smelled fresh, but only Shtyrkov was eating. Gibson’s face twisted in tension, but only for a moment. Svetlana poured him a coffee.