Optimal paths on the road network as directed polymers

Transcription

1 1/12 Optimal paths on the road network as directed polymers Alex Solon, Guy Bunin, Sherry Chu, Mehran Kardar [arxiv: ]

2 2/12 Directed polymer in a random medium (DPRM) Statistics of a stretched chain in a disordered environment Fracture line of a torn paper sheet Domain wall in a 2d Ising model with random bonds Vortex lines in superconductors

3 2/12 Directed polymer in a random medium (DPRM) Statistics of a stretched chain in a disordered environment Fracture line of a torn paper sheet Domain wall in a 2d Ising model with random bonds Vortex lines in superconductors Mapping to: the fluctuations of a growing interface (KPZ equation) the noisy Burger equation

4 DPRM scaling 3/12 h(x) 0 d x E[h(x)] = d 0 ( ) P[h] exp 1 k B T E[h] [ γ 2 ( dh ) 2 ] dx + V (x, h) dx Random potential

5 DPRM scaling 3/12 h(x) 0 d x E[h(x)] = d 0 (E Ē)2 d 2β ( ) P[h] exp 1 k B T E[h] [ γ 2 ( dh ) 2 ] dx + V (x, h) dx Random potential (h h) 2 x 2ζ (1+1)d: β = 1/3, ζ = 2/3 Short-range disorder

6 DPRM scaling h(x) 0 d x E[h(x)] = d 0 (E Ē)2 d 2β ( ) P[h] exp 1 k B T E[h] [ γ 2 ( dh ) 2 ] dx + V (x, h) dx Random potential (h h) 2 x 2ζ (1+1)d: β = 1/3, ζ = 2/3 Short-range disorder 10 0 σp (E) 10 2 Universal fluctuations DPRM Gaussian DPRM Exponential Tracy-Widom 10 8 (E E )/σ /12

7 Optimal path on the road network 4/12 DPRM at T = 0 = Configuration minimizing energy

8 Optimal path on the road network 4/12 DPRM at T = 0 = Configuration minimizing energy

9 Optimal path on the road network 4/12 DPRM at T = 0 = Configuration minimizing energy

10 Optimal path on the road network 4/12 DPRM at T = 0 = Configuration minimizing energy Fastest route by car Shortest route in distance Travel time or distance Energy of a DPRM

11 Questions 5/12 Do optimal paths follow universal scaling laws? = V (x)?

12 5/12 Questions Do optimal paths follow universal scaling laws? = V (x)? What does P(V ) look like? Look at short paths Can we forget about the details of the network on large scale? Increase distance

13 Fractal structure 6/12 DPRM model [Halpin-Healy and Zhang, Physics Reports 95] Shortest path from Munich d = 300km

14 Short paths 7/12 Three data sets Open Street Map Open Source Routing Machine

15 7/12 Short paths Three data sets Open Street Map Open Source Routing Machine Paths between points at distance d = 1km P (L) Shortest path Europe US Asia L L(km)

16 Short paths 7/12 Three data sets Open Street Map Open Source Routing Machine Paths between points at distance d = 1km P (L) Shortest path L 3 L(km) Europe US Asia P (T ) Universal scaling at short distance! Fastest path Europe US Asia 10 5 T T (s)

17 8/12 Larger distances P (L d) Pm d = 3km d = 10km 0.2 d = 30km d = 100km β = 0.66 Tracy-Widom 0.0 (L L m )/d β

18 Larger distances P (L d) Pm d = 3km d = 10km 0.2 d = 30km d = 100km β = 0.66 Tracy-Widom 0.0 (L L m )/d β DPRM model with power-law noise P (E d) Pm d = 30 d = 100 d = 300 d = 1000 Tracy-Widom β = 1/3 (E E m )/d β /12

19 Larger distances P (L d) Pm d = 3km d = 10km 0.2 d = 30km d = 100km β = 0.66 Tracy-Widom 0.0 (L L m )/d β DPRM model with power-law noise P (E d) Pm d = 30 d = 100 d = 300 d = 1000 Tracy-Widom β = 1/3 (E E m )/d β same distributions Different exponents β [0.58, 0.75] 8/12

20 Transverse deviation 9/ h(x) a(km) Europe, ζ = 0.72 US, ζ = 0.75 Asia, ζ = 0.69 h x ζ fixed d = 1000km 10 1 x(km) ζ > 2/3

21 Transverse deviation 9/ h(x) a(km) Europe, ζ = 0.72 US, ζ = 0.75 Asia, ζ = 0.69 h x ζ fixed d = 1000km 10 1 x(km) ζ > 2/3 Can we explain the different scaling exponents?

22 Long-range correlations - DPRM 10/12 LR correlations in the noise change the exponents η(x, y)η(x, y ) = y y 2ρ 1 δ(x x ) [Chu and Kardar, PRE 2016]

23 Long-range correlations - Road density Look at correlations of road density Discretized maps C(r ) = hρ(x + r )ρ(x)i hρi2 11/12

24 Long-range correlations - Road density Look at correlations of road density Discretized maps C(r ) = hρ(x + r )ρ(x)i hρi C(r) r Slow decay Europe US Asia r 0.5 r(km) /12

25 Conclusion 12/12 Universal power-law distribution at short scale Scaling laws Convergence to universal distribution on larger scale Non-universal exponents due to LR correlations