Distinct and Common Sites visited by N
random Walkers
Satya N. Majumdar
Laboratoire de Physique Théorique et Modèles Statistiques,CNRS,
Université Paris-Sud, France
Collaborators:
A. Kundu (LPTMS, Orsay, FRANCE)
G. Schehr (LPTMS, Orsay, FRANCE)
M. V. Tamm (Moscow University, Russia)
Refs:
• S.M. and M. V. Tamm, Phys. Rev. E 86, 021135 (2012)
• A. Kundu, S.M. and G. Schehr, Phys. Rev. Lett. 110, 220602 (2013)
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Plan:
• N independent random walkers, each of t-steps, moving on a
d-dimensional regular lattice
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Plan:
• N independent random walkers, each of t-steps, moving on a
d-dimensional regular lattice
• Two objects of interest
DN (t) → no. of distinct sites visited by N walkers
CN (t) → no. of common sites visited by N walkers
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Plan:
• N independent random walkers, each of t-steps, moving on a
d-dimensional regular lattice
• Two objects of interest
DN (t) → no. of distinct sites visited by N walkers
CN (t) → no. of common sites visited by N walkers
• Interesting phase transition in the asymptotic growth of hCN (t)i
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Plan:
• N independent random walkers, each of t-steps, moving on a
d-dimensional regular lattice
• Two objects of interest
DN (t) → no. of distinct sites visited by N walkers
CN (t) → no. of common sites visited by N walkers
• Interesting phase transition in the asymptotic growth of hCN (t)i
• Exact distributions of DN (t) and CN (t) in d = 1
=⇒ link to Extreme Value Statistics
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Plan:
• N independent random walkers, each of t-steps, moving on a
d-dimensional regular lattice
• Two objects of interest
DN (t) → no. of distinct sites visited by N walkers
CN (t) → no. of common sites visited by N walkers
• Interesting phase transition in the asymptotic growth of hCN (t)i
• Exact distributions of DN (t) and CN (t) in d = 1
=⇒ link to Extreme Value Statistics
• Summary and Conclusion
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Distinct sites visited by a single random walker
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D1 (t) → no. of distinct sites visited
by a RW of t steps
→ sites visited atleast once
physics, chemistry, metallurgy, ecology,...
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Distinct sites visited by a single random walker
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D1 (t) → no. of distinct sites visited
by a RW of t steps
→ sites visited atleast once
physics, chemistry, metallurgy, ecology,...
For large t
Random walk
√ d
t ,
d <2
recurrent for d < 2
∼ t/ln t,
d =2
transient for d > 2
∼ γd t,
d >2
hD1 (t)i ∼
[Dvoretzky & Erdös (’51), Vineyard (’63), Montrol & Weiss (’65), .....]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Distinct sites visited by N indep. random walkers
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DN (t) → no. of distinct sites visited by N indep. walkers each of t steps
→ sites visited atleast once by any of the walkers
[ Larralde, Trunfio, Havlin, Stanley, & Weiss, Nature, 355, 423 (1992)]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Number of distinct sites: growth with time t
D N (t)
I
0
vs.
t
III
II
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t*1
t2*
t
Asymptotic late time (t >> t2∗ ) growth:
√ d
hDN (t)i ∼ (ln N)d/2
t
d <2
∼ N t/ln t
d =2
∼ Nt
d >2
[Larralde et.
S.N. Majumdar
al. (’92)]
Distinct and Common Sites visited by N random Walkers
Union and Intersection of the visited sites
Si (t) → the set of sites visited by the i-th walker up to t
S1
S2
S1
S3
S2
S3
Distinct sites DN (t) ≡ size of S1 (t)∪S2 (t)∪S3 (t) . . . ∪SN (t) → Union
A natural counterpart:
Common sites CN (t) ≡ size of S1 (t)∩S2 (t)∩S3 (t) . . . ∩SN (t)
→ Intersection
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Common sites visited by N indep. random walkers
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CN (t) → no. of common sites visited by N indep. walkers each of t steps
→ sites visited by all the walkers (green sites)
[S.M. & M. V. Tamm, Phys.
S.N. Majumdar
Rev. E, 86, 021135 (2012)]
Distinct and Common Sites visited by N random Walkers
Common Sites
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light
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Common Sites
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light
light transmits from bottom to top provided the same site in each of the N
planes is visited by the corresponding walker
Intensity of transmitted light IN (t) ∝ CN (t)
−→ no. of common sites visited by N independent walkers in a single plane
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Common Sites
000
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light
light transmits from bottom to top provided the same site in each of the N
planes is visited by the corresponding walker
Intensity of transmitted light IN (t) ∝ CN (t)
−→ no. of common sites visited by N independent walkers in a single plane
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Asymptotic growth of hCN (t)i
III
d
d c = (2N−N1)
II
2
0
I
1

bN t d/2








t/(ln t)N





hCN (t)i ∼
t ν=N−d(N−1)/2






ln t







const.
d <2
d =2
2 < d < dc (N)
d = dc (N)
d > dc (N)
N
[S.M. & M. V. Tamm, Phys.
Rev. E, 86, 021135 (2012)]
Nontrivial exponent ν = N − d(N − 1)/2 in the intermediate Phase II
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Asymptotic growth of hCN (t)i
III
d
d c = (2N−N1)
II
2
0
I
1

bN t d/2








t/(ln t)N





hCN (t)i ∼
t ν=N−d(N−1)/2






ln t







const.
d <2
d =2
2 < d < dc (N)
d = dc (N)
d > dc (N)
N
[S.M. & M. V. Tamm, Phys.
Rev. E, 86, 021135 (2012)]
Nontrivial exponent ν = N − d(N − 1)/2 in the intermediate Phase II
Few Examples:
• N = 2 → dc = 4;
hC2 (t)i ∼ t 1/2
S.N. Majumdar
in 2 < d = 3 < 4
Distinct and Common Sites visited by N random Walkers
Asymptotic growth of hCN (t)i
III
d
d c = (2N−N1)
II
2
0
I
1

bN t d/2








t/(ln t)N





hCN (t)i ∼
t ν=N−d(N−1)/2






ln t







const.
d <2
d =2
2 < d < dc (N)
d = dc (N)
d > dc (N)
N
[S.M. & M. V. Tamm, Phys.
Rev. E, 86, 021135 (2012)]
Nontrivial exponent ν = N − d(N − 1)/2 in the intermediate Phase II
Few Examples:
• N = 2 → dc = 4;
hC2 (t)i ∼ t 1/2
in 2 < d = 3 < 4
• N = 3 → dc = 3;
hC3 (t)i ∼ ln t
in d = 3
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Asymptotic growth of hCN (t)i
III
d
d c = (2N−N1)
II
2
0
I
1

bN t d/2








t/(ln t)N





hCN (t)i ∼
t ν=N−d(N−1)/2






ln t







const.
d <2
d =2
2 < d < dc (N)
d = dc (N)
d > dc (N)
N
[S.M. & M. V. Tamm, Phys.
Rev. E, 86, 021135 (2012)]
Nontrivial exponent ν = N − d(N − 1)/2 in the intermediate Phase II
Few Examples:
• N = 2 → dc = 4;
hC2 (t)i ∼ t 1/2
in 2 < d = 3 < 4
• N = 3 → dc = 3;
hC3 (t)i ∼ ln t
in d = 3
• N = 4 → dc = 2;
hC4 (t)i ∼ t/(ln t)4
in d = 2
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Heuristic scaling argument
• In time t, a single walker explores
a volume V (t) ∼ t d/2
t
0
• Fraction of sites visited by the single
(t)i
walker: φ = hDV1(t)
→ prob. that a site is visited by the walker
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Heuristic scaling argument
• In time t, a single walker explores
a volume V (t) ∼ t d/2
t
0
• Fraction of sites visited by the single
(t)i
walker: φ = hDV1(t)
→ prob. that a site is visited by the walker
For N indep. walkers, prob. that a site is visited by all ∼ φN
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Heuristic scaling argument
• In time t, a single walker explores
a volume V (t) ∼ t d/2
t
0
• Fraction of sites visited by the single
(t)i
walker: φ = hDV1(t)
→ prob. that a site is visited by the walker
For N indep. walkers, prob. that a site is visited by all ∼ φN
h
iN
1 (t)i
⇒ hCN (t)i ∼ φN V (t) ∼ hDt d/2
t d/2
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Heuristic scaling argument
• In time t, a single walker explores
a volume V (t) ∼ t d/2
t
0
• Fraction of sites visited by the single
(t)i
walker: φ = hDV1(t)
→ prob. that a site is visited by the walker
For N indep. walkers, prob. that a site is visited by all ∼ φN
h
iN
1 (t)i
⇒ hCN (t)i ∼ φN V (t) ∼ hDt d/2
t d/2
•d <2 :
hD1 (t)i ∼ t d/2 =⇒ hCN (t)i ∼ t d/2
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Heuristic scaling argument
• In time t, a single walker explores
a volume V (t) ∼ t d/2
t
• Fraction of sites visited by the single
(t)i
walker: φ = hDV1(t)
0
→ prob. that a site is visited by the walker
For N indep. walkers, prob. that a site is visited by all ∼ φN
h
iN
1 (t)i
⇒ hCN (t)i ∼ φN V (t) ∼ hDt d/2
t d/2
•d <2 :
hD1 (t)i ∼ t d/2 =⇒ hCN (t)i ∼ t d/2
•d >2 :
hD1 (t)i ∼ t
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Heuristic scaling argument
• In time t, a single walker explores
a volume V (t) ∼ t d/2
t
• Fraction of sites visited by the single
(t)i
walker: φ = hDV1(t)
0
→ prob. that a site is visited by the walker
For N indep. walkers, prob. that a site is visited by all ∼ φN
h
iN
1 (t)i
⇒ hCN (t)i ∼ φN V (t) ∼ hDt d/2
t d/2
•d <2 :
hD1 (t)i ∼ t d/2 =⇒ hCN (t)i ∼ t d/2
•d >2 :
hD1 (t)i ∼ t
=⇒ hCN (t)i ∼ t ν=N−(N−1)d/2 if ν > 0, i.e., for d < dc = 2N/(N − 1)
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Heuristic scaling argument
• In time t, a single walker explores
a volume V (t) ∼ t d/2
t
• Fraction of sites visited by the single
(t)i
walker: φ = hDV1(t)
0
→ prob. that a site is visited by the walker
For N indep. walkers, prob. that a site is visited by all ∼ φN
h
iN
1 (t)i
⇒ hCN (t)i ∼ φN V (t) ∼ hDt d/2
t d/2
•d <2 :
hD1 (t)i ∼ t d/2 =⇒ hCN (t)i ∼ t d/2
•d >2 :
hD1 (t)i ∼ t
=⇒ hCN (t)i ∼ t ν=N−(N−1)d/2 if ν > 0, i.e., for d < dc = 2N/(N − 1)
∼ const.
if ν < 0, i.e., for d > dc
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Exact computation
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σ(~x , t) = 1 if ~x visited by all t-step walkers (green)
= 0 otherwise
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Exact computation
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σ(~x , t) = 1 if ~x visited by all t-step walkers (green)
= 0 otherwise
X
Then CN (t) =
σ(~x , t)
~
x
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Exact computation
O
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σ(~x , t) = 1 if ~x visited by all t-step walkers (green)
= 0 otherwise
X
Then CN (t) =
σ(~x , t)
~
x
•
hCN (t)i =
X
X
N
hσ(~x , t)i =
[p(~x , t)]
~
x
~
x
p(~x , t) → prob. that ~x is visited by a single t-step walker
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Exact computation
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σ(~x , t) = 1 if ~x visited by all t-step walkers (green)
= 0 otherwise
X
Then CN (t) =
σ(~x , t)
~
x
•
hCN (t)i =
X
X
N
hσ(~x , t)i =
[p(~x , t)]
~
x
~
x
p(~x , t) → prob. that ~x is visited by a single t-step walker
N
• Also [1 − p(~x , t)] → prob. that ~x is NOT visited by any of the walkers
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Exact computation
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σ(~x , t) = 1 if ~x visited by all t-step walkers (green)
= 0 otherwise
X
Then CN (t) =
σ(~x , t)
~
x
•
hCN (t)i =
X
X
N
hσ(~x , t)i =
[p(~x , t)]
~
x
~
x
p(~x , t) → prob. that ~x is visited by a single t-step walker
N
• Also [1 − p(~x , t)] → prob. that ~x is NOT visited by any of the walkers
i
Xh
N
=⇒ hDN (t)i =
1 − [1 − p(~x , t)]
~
x
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Exact computation
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σ(~x , t) = 1 if ~x visited by all t-step walkers (green)
= 0 otherwise
X
Then CN (t) =
σ(~x , t)
~
x
•
hCN (t)i =
X
X
N
hσ(~x , t)i =
[p(~x , t)]
~
x
~
x
p(~x , t) → prob. that ~x is visited by a single t-step walker
N
• Also [1 − p(~x , t)] → prob. that ~x is NOT visited by any of the walkers
i
Xh
N
=⇒ hDN (t)i =
1 − [1 − p(~x , t)]
~
x
p(~x , t) −→ central quantity
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
The probability p(~x , t)
p(~x , t) → prob. that ~x is visited by a single t-step walker
Let τ −→ last time before t the site ~x
was visited
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t _ step random walker
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
The probability p(~x , t)
p(~x , t) → prob. that ~x is visited by a single t-step walker
Let τ −→ last time before t the site ~x
was visited
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x
Z
o
t
G (~x , τ ) q(t − τ ) dτ
Then p(~x , t) =
0
t _ step random walker
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
The probability p(~x , t)
p(~x , t) → prob. that ~x is visited by a single t-step walker
Let τ −→ last time before t the site ~x
was visited
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x
Z
o
t
G (~x , τ ) q(t − τ ) dτ
Then p(~x , t) =
0
t _ step random walker
• G (~x , t) = (4πDτ )−d/2 exp −x 2 /(4Dτ ) → diffusion Green’s function
• q(τ ) → prob. of no return to the starting pt. in time τ
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Scaling form of p(~x , t):


√ x
f

<

4πD t





1
p(~x , t) ∼
f2 √4 πx D t
ln
t






 1−d/2

t
f> √4 πx D t
(d < 2)
(d = 2)
(d > 2)
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Scaling form of p(~x , t):


√ x
f

<

4πD t





1
p(~x , t) ∼
f2 √4 πx D t
ln
t






 1−d/2

t
f> √4 πx D t
(d < 2)
(d = 2)
f< (z) ∼ const.
∼z
−d
e
as z → 0
−z 2
f> (z) ∼ z 2−d
(d > 2)
S.N. Majumdar
∼ z −2 e −z
as z → ∞
as z → 0
2
as z → ∞
Distinct and Common Sites visited by N random Walkers
Scaling form of p(~x , t):


√ x
f

<

4πD t





1
p(~x , t) ∼
f2 √4 πx D t
ln
t






 1−d/2

t
f> √4 πx D t
Z
hCN (t)i =
(d < 2)
f< (z) ∼ const.
∼z
(d = 2)
−d
e
as z → 0
−z 2
f> (z) ∼ z 2−d
(d > 2)
N
Z
∼ z −2 e −z
∞
d~x [p(~x , t)] ∼
dx x d−1 [p(~x , t)]
N
as z → ∞
as z → 0
2
as z → ∞
=⇒
1
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Exact asymptotic results
III
d
d c = (2N−N1)
II
2
0
I
1

bN t d/2








t/(ln t)N





hCN (t)i ∼
t ν=N−d(N−1)/2






ln t







const.
d <2
d =2
2 < d < dc (N)
d = dc (N)
d > dc (N)
N
[S.M. & M. V. Tamm, Phys.
S.N. Majumdar
Rev. E, 86, 021135 (2012)]
Distinct and Common Sites visited by N random Walkers
Distribution of Distinct and Common sites
O
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000
000
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000
111
000
111
DN (t), CN (t) → no. of
distinct/common sites visited by N
indep. walkers each of t steps
DN (t) and CN (t) → random variables
What about the full distribution of DN (t) and CN (t) ?
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Distribution of Distinct and Common sites
O
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000
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000
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000
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111
000
000
111
000
111
000
111
DN (t), CN (t) → no. of
distinct/common sites visited by N
indep. walkers each of t steps
DN (t) and CN (t) → random variables
What about the full distribution of DN (t) and CN (t) ?
Focus on d = 1:
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Distribution of Distinct and Common sites
O
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000
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000
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000
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000
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00
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111
000
000
111
000
111
000
111
DN (t), CN (t) → no. of
distinct/common sites visited by N
indep. walkers each of t steps
DN (t) and CN (t) → random variables
What about the full distribution of DN (t) and CN (t) ?
Focus on d = 1:
• maximum overlap between walkers in d = 1 −→ nontrivial
• interesting link to Extreme Value Statistics −→ exactly solvable
• Various applications in d = 1:
biological applications ⇒ proteins diffusing along DNA
environmental applications ⇒ diffusion of river pollutants
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
A single walker N = 1 in one dimension
D1 (t) = C1 (t) = D1+ (t) + D1− (t)
−→ span of the walk
1
0
1
0
1
0
0
1
1
0
0
1
M1
1
0
1
0
O
where
1
0
1
0
1
0
0
1
1
0
1
0
m1
time
D1+ (t) = M1 (t) = max {x1 (τ )}
0≤τ ≤t
D1− (t) = m1 (t) = − min {x1 (τ )}
1
0
0
1
0≤τ ≤t
=⇒ link to Extreme Value Statistics
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
A single walker N = 1 in one dimension
D1 (t) = C1 (t) = D1+ (t) + D1− (t)
−→ span of the walk
1
0
1
0
1
0
0
1
1
0
0
1
M1
1
0
1
0
O
where
1
0
1
0
1
0
0
1
1
0
1
0
m1
time
D1+ (t) = M1 (t) = max {x1 (τ )}
0≤τ ≤t
D1− (t) = m1 (t) = − min {x1 (τ )}
1
0
0
1
0≤τ ≤t
=⇒ link to Extreme Value Statistics
Note that {M1 (t), m1 (t)} −→ correlated random variables
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Joint distribution of M1 (t) and m1 (t)
L1
M1
O
1
0
1
0
m1
time
L2
S.N. Majumdar
Prob.[M1 (t) ≤ L1 , m1 (t) ≤ L2 ]
−→ prob. that the walker stays inside
the box [−L2 , L1 ] up to time t
−−−→ g √4L1D t = l1 , √4L2D t = l2
t→∞
Distinct and Common Sites visited by N random Walkers
space
Joint distribution of M1 (t) and m1 (t)
L1
M1
O
1
0
1
0
m1
time
L2
Prob.[M1 (t) ≤ L1 , m1 (t) ≤ L2 ]
−→ prob. that the walker stays inside
the box [−L2 , L1 ] up to time t
−−−→ g √4L1D t = l1 , √4L2D t = l2
t→∞
Solving Fokker-Planck equation with absorbing b.c. at L1 and −L2 gives
∞
4X 1
(2n + 1)πl2
(2n + 1)2 π 2
g (l1 , l2 ) =
sin
exp −
π n=0 2n + 1
l1 + l2
4(l1 + l2 )2
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Prob. density of distinct/common sites for N = 1
Joint distribution:
Prob.[M1 (t) ≤ L1 , m1 (t) ≤ L2 ] → g √4L1D t = l1 , √4L2D t = l2
No. of distinct/common sites D1 (t) = C1 (t) = M1 (t) + m1 (t)
D1 (t)
1
Prob. density: Prob.[D1 (t)] = Prob.[C1 (t)] → √4Dt
P1 √
4Dt
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Prob. density of distinct/common sites for N = 1
Joint distribution:
Prob.[M1 (t) ≤ L1 , m1 (t) ≤ L2 ] → g √4L1D t = l1 , √4L2D t = l2
No. of distinct/common sites D1 (t) = C1 (t) = M1 (t) + m1 (t)
D1 (t)
1
Prob. density: Prob.[D1 (t)] = Prob.[C1 (t)] → √4Dt
P1 √
4Dt
Z
∞
Z
P1 (x) =
0
0
∞
∂2g
δ(x − l1 − l2 ) dl1 dl2
∂l1 ∂l2
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Prob. density of distinct/common sites for N = 1
Joint distribution:
Prob.[M1 (t) ≤ L1 , m1 (t) ≤ L2 ] → g √4L1D t = l1 , √4L2D t = l2
No. of distinct/common sites D1 (t) = C1 (t) = M1 (t) + m1 (t)
D1 (t)
1
Prob. density: Prob.[D1 (t)] = Prob.[C1 (t)] → √4Dt
P1 √
4Dt
Z
∞
Z
P1 (x) =
0
0
∞
∂2g
δ(x − l1 − l2 ) dl1 dl2
∂l1 ∂l2
∞
2 2
8 X
P1 (x) = √
(−1)m+1 m2 e −m x
π m=1
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Prob. density of distinct/common sites for N = 1
Joint distribution:
Prob.[M1 (t) ≤ L1 , m1 (t) ≤ L2 ] → g √4L1D t = l1 , √4L2D t = l2
No. of distinct/common sites D1 (t) = C1 (t) = M1 (t) + m1 (t)
D1 (t)
1
Prob. density: Prob.[D1 (t)] = Prob.[C1 (t)] → √4Dt
P1 √
4Dt
Z
∞
Z
∞
P1 (x) =
0
0
∂2g
δ(x − l1 − l2 ) dl1 dl2
∂l1 ∂l2
∞
2 2
8 X
P1 (x) = √
(−1)m+1 m2 e −m x
π m=1
1.5
N=1


√8
π
e −x
2
x →0
x →∞
1
P1(x)
P1 (x) →

2 −5 −π 2 /4 x 2

 2π x e
0.5
0
0
1
2
x
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
3
space
Multiple (N > 1) t-step walkers
1
0
0
1
0
1
0
1
1
0
0
1
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
1
0
0
1
1
0
1
0
1
0
0
1
1
0
0
1
m2
time
m1
1
0
0
1
1
0
1
0
Mi (t) → maximum of the i-th walker
−mi (t) → minimum of the i-th walker
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distinct sites DN (t) for (N > 1) walkers
1
0
0
1
1
0
1
0
1
0
1
0
+
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
DN
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
D −N
1
0
1
0
Mi (t) → maximum of the i-th
walker
−mi (t) → minimum of the i-th
walker
1
0
0
1
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distinct sites DN (t) for (N > 1) walkers
1
0
0
1
1
0
1
0
1
0
1
0
+
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
DN
1
0
1
0
1
0
1
0
1
0
1
0
m2
1
0
1
0
m1
time
D −N
1
0
1
0
Mi (t) → maximum of the i-th
walker
−mi (t) → minimum of the i-th
walker
1
0
0
1
Distinct:
DN (t) = DN+ (t) + DN− (t) where
DN+ (t) = max [M1 (t), M2 (t), . . . , MN (t)]
DN− (t) = max [m1 (t), m2 (t), . . . , mN (t)]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Common sites CN (t) for (N > 1) walkers
1
0
0
1
0
1
0
1
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
0
1
O
M1
C+N
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
m2
m1
time
C N−
1
0
1
0
Mi (t) → maximum of the i-th
walker
−mi (t) → minimum of the i-th
walker
1
0
1
0
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Common sites CN (t) for (N > 1) walkers
1
0
0
1
0
1
0
1
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
0
1
O
M1
C+N
1
0
0
1
1
0
0
1
1
0
1
0
m2
1
0
1
0
m1
time
C N−
1
0
1
0
Mi (t) → maximum of the i-th
walker
−mi (t) → minimum of the i-th
walker
1
0
1
0
Common:
CN (t) = CN+ (t) + CN− (t) where
CN+ (t) = min [M1 (t), M2 (t), . . . , MN (t)]
CN− (t) = min [m1 (t), m2 (t), . . . , mN (t)]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distribution of DN (t)
1
0
0
1
1
0
1
0
1
0
1
0
+
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
DN
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
−
DN
1
0
1
0
DN (t) = DN+ (t) + DN− (t)
DN+ (t) = max {Mi (t)}
1≤i≤N
DN− (t) = max {mi (t)}
1
0
0
1
1≤i≤N
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distribution of DN (t)
1
0
0
1
1
0
1
0
1
0
1
0
M2
1
0
1
0
O
DN (t) = DN+ (t) + DN− (t)
+
1
0
0
1
1
0
0
1
M1
DN
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
DN+ (t) = max {Mi (t)}
1≤i≤N
−
DN
1
0
1
0
DN− (t) = max {mi (t)}
1
0
0
1
1≤i≤N
N
Y
Prob. [Mi (t) ≤ L1 , mi (t) ≤ L2 ]
Prob. DN+ (t) ≤ L1 , DN− (t) ≤ L2 =
i=1
N
= [g (l1 , l2 )] ;
S.N. Majumdar
l1 =
√L1 ,
4Dt
l2 =
√L2
4Dt
Distinct and Common Sites visited by N random Walkers
space
Distribution of DN (t)
1
0
0
1
1
0
1
0
1
0
1
0
M2
1
0
1
0
O
DN (t) = DN+ (t) + DN− (t)
+
1
0
0
1
1
0
0
1
M1
DN
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
DN+ (t) = max {Mi (t)}
1≤i≤N
−
DN
1
0
1
0
DN− (t) = max {mi (t)}
1
0
0
1
1≤i≤N
N
Y
Prob. [Mi (t) ≤ L1 , mi (t) ≤ L2 ]
Prob. DN+ (t) ≤ L1 , DN− (t) ≤ L2 =
i=1
N
= [g (l1 , l2 )] ;
(t)
1
√N
Prob. density: Prob.[DN (t)] → √4Dt
PN D
4Dt
S.N. Majumdar
l1 =
√L1 ,
4Dt
l2 =
√L2
4Dt
Distinct and Common Sites visited by N random Walkers
space
Distribution of DN (t)
1
0
0
1
1
0
1
0
1
0
1
0
M2
1
0
1
0
O
DN (t) = DN+ (t) + DN− (t)
+
1
0
0
1
1
0
0
1
DN
M1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
time
m1
DN+ (t) = max {Mi (t)}
1≤i≤N
−
DN
1
0
1
0
DN− (t) = max {mi (t)}
1
0
0
1
1≤i≤N
N
Y
Prob. [Mi (t) ≤ L1 , mi (t) ≤ L2 ]
Prob. DN+ (t) ≤ L1 , DN− (t) ≤ L2 =
i=1
N
= [g (l1 , l2 )] ;
(t)
1
√N
Prob. density: Prob.[DN (t)] → √4Dt
PN D
4Dt
PN (x) =
R∞R∞
0
0
∂2g N
∂l1 ∂l2
l1 =
√L1 ,
4Dt
l2 =
√L2
4Dt
δ(x − l1 − l2 ) dl1 dl2
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distribution of CN (t)
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
C+N
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
C−
CN (t) = CN+ (t) + CN− (t)
CN+ (t) = min {Mi (t)}
1≤i≤N
N
1
0
1
0
CN− (t) = min {mi (t)}
1
0
0
1
1≤i≤N
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distribution of CN (t)
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
CN (t) = CN+ (t) + CN− (t)
C+N
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
CN+ (t) = min {Mi (t)}
time
C−
1≤i≤N
N
1
0
1
0
CN− (t) = min {mi (t)}
1
0
0
1
1≤i≤N
N
Y
Prob. [Mi (t) ≥ L1 , mi (t) ≥ L2 ]
Prob. CN+ (t) ≥ L1 , CN− (t) ≥ L2 =
i=1
N
= [h (l1 , l2 )]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distribution of CN (t)
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
CN (t) = CN+ (t) + CN− (t)
C+N
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
CN+ (t) = min {Mi (t)}
time
C−
1≤i≤N
N
1
0
1
0
CN− (t) = min {mi (t)}
1
0
0
1
1≤i≤N
N
Y
Prob. [Mi (t) ≥ L1 , mi (t) ≥ L2 ]
Prob. CN+ (t) ≥ L1 , CN− (t) ≥ L2 =
i=1
N
= [h (l1 , l2 )]
where h(l1 , l2 ) = 1 − erf(l1 ) − erf(l2 ) + g (l1 , l2 )
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Distribution of CN (t)
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
CN (t) = CN+ (t) + CN− (t)
C+N
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
CN+ (t) = min {Mi (t)}
time
C−
1≤i≤N
N
1
0
1
0
CN− (t) = min {mi (t)}
1
0
0
1
1≤i≤N
N
Y
Prob. [Mi (t) ≥ L1 , mi (t) ≥ L2 ]
Prob. CN+ (t) ≥ L1 , CN− (t) ≥ L2 =
i=1
N
= [h (l1 , l2 )]
where h(l1 , l2 ) = 1 − erf(l1 ) − erf(l2 ) + g (l1 , l2 )
CN (t)
1
Prob. density: Prob.[CN (t)] → √4Dt
QN √
4Dt
QN (x) =
R∞R∞
0
S.N. Majumdar
0
∂ 2 hN
∂l1 ∂l2
δ(x − l1 − l2 ) dl1 dl2
Distinct and Common Sites visited by N random Walkers
Distributions for fixed N
1.5
P2(x), Q2(x)
N=2 : Distinct
N=2 : Common
1.0
0.5
0.0
0
1
2
3
4
x
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Distributions for fixed N
1.5
P2(x), Q2(x)
N=2 : Distinct
N=2 : Common
1.0
0.5
0.0
0
1
2
3
4
x
Asymptotics for fixed N ≥ 2:
Distinct:
PN (x) →
Common:

2
2
 aN x −5 e −N π /4 x

bN e −x
2
/2
x →0
QN (x) →
x →∞

 cN x

dN x 1−N e −N x
x →0
2
x →∞
[Kundu, S.M. & Schehr, PRL, 110, 220602 (2013)]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
First and second moments for large N:
First moment:
Distinct:
hDN (t)i
√
4Dt
=2
R∞
Common:
hCN (t)i
√
4Dt
=2
R∞
0
0
x
d
N
dx [erf(x)]
dx
[erfc(x)]N dx
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
First and second moments for large N:
First moment:
Distinct:
hDN (t)i
√
4Dt
=2
R∞
Common:
hCN (t)i
√
4Dt
=2
R∞
0
0
x
d
N
dx [erf(x)]
dx
[erfc(x)]N dx
Large N:
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
First and second moments for large N:
First moment:
Distinct:
hDN (t)i
√
4Dt
=2
R∞
Common:
hCN (t)i
√
4Dt
=2
R∞
0
0
x
d
N
dx [erf(x)]
dx
[erfc(x)]N dx
Large N:
Distinct:
hDN (t)i
√
4Dt
Var
h
DN (t)
√
4Dt
i
√
≈ 2 ln N +
≈
√γ ;
ln N
2α
ln N ;
γ = 0.577216... → Euler const.
α = γ + π 2 /6
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
First and second moments for large N:
First moment:
Distinct:
hDN (t)i
√
4Dt
=2
R∞
Common:
hCN (t)i
√
4Dt
=2
R∞
0
0
x
d
N
dx [erf(x)]
dx
[erfc(x)]N dx
Large N:
Distinct:
hDN (t)i
√
4Dt
Var
h
DN (t)
√
4Dt
i
√
≈ 2 ln N +
≈
√γ ;
ln N
2α
ln N ;
γ = 0.577216... → Euler const.
α = γ + π 2 /6
Common:
hCN (t)i
√
4Dt
Var
h
C (t)
√N
4Dt
i
√
≈
π
N
≈
π
2N 2
decreases with N
S.N. Majumdar
!!
Distinct and Common Sites visited by N random Walkers
Scaling form for the distribution of DN (t)
N
PN(x)
peak position
peak width
ln N
~
~ 1/ ln N
Suggests a scaling form:
√
√
√
PN (x) ∼ 2 ln N D 2 ln N x − 2 ln N
x
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Scaling form for the distribution of DN (t)
N
peak position
PN(x)
peak width
ln N
~
~ 1/ ln N
Suggests a scaling form:
√
√
√
PN (x) ∼ 2 ln N D 2 ln N x − 2 ln N
x
Exact scaling analysis gives:
D(y ) = 2 e −y K0 2 e −y /2 where K0 (z) → modified Bessel function
D(y ) →

 y e −y
y →∞
 √
y → −∞
π e −3y /4 exp −2 e −y /2
[Kundu, S.M. & Schehr, PRL, 110, 220602 (2013)]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Simple interpretation of the scaling function
1
0
0
1
1
0
1
0
1
0
0
1
+
1
0
0
1
1
0
1
0
M2
1
0
1
0
O
M1
DN
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
−
DN+ (t) = max {Mi (t)}
1≤i≤N
DN
1
0
1
0
DN− (t) = max {mi (t)}
1
0
0
1
√Mi
4Dt
DN (t) = DN+ (t) + DN− (t)
1≤i≤N
= zi ’s → i.i.d variables each drawn from: p(z) =
√2
π
2
e −z with z ≥ 0
⇒ DN+ (centered and scaled) → Gumbel distributed:
PG (x) = e −x exp [−e −x ]
As N → ∞, DN+ and DN− becomes uncorrelated
Thus, DN (t) → sum of two independent Gumbel variables
R∞
D(y ) = −∞ dx PG (x) PG (y − x) = 2 e −y K0 2 e −y /2
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Scaling form for the distribution of CN (t)
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
C+N
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
C−
CN (t) = CN+ (t) + CN− (t)
CN+ (t) = min {Mi (t)}
1≤i≤N
N
1
0
1
0
CN− (t) = min {mi (t)}
1
0
0
1
1≤i≤N
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Scaling form for the distribution of CN (t)
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
C+N
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
C−
CN (t) = CN+ (t) + CN− (t)
CN+ (t) = min {Mi (t)}
1≤i≤N
N
1
0
1
0
CN− (t) = min {mi (t)}
1
0
0
1
1≤i≤N
Exact large N analysis gives: QN (x) ∼ N C(N x) where
h
i
C(y ) = π4 y exp − √2π y
[Kundu, S.M. & Schehr, PRL, 110, 220602 (2013)]
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
space
Scaling form for the distribution of CN (t)
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
M2
1
0
1
0
O
M1
C+N
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
m2
m1
time
C−
CN (t) = CN+ (t) + CN− (t)
CN+ (t) = min {Mi (t)}
1≤i≤N
N
1
0
1
0
CN− (t) = min {mi (t)}
1
0
0
1
1≤i≤N
Exact large N analysis gives: QN (x) ∼ N C(N x) where
h
i
C(y ) = π4 y exp − √2π y
[Kundu, S.M. & Schehr, PRL, 110, 220602 (2013)]
Interpretation: For large N, CN+ and CN− get uncorrelated
Thus, CN (t) → sum of two independent Weibull variables√
each distributed with PW (x) = √2π e −2x/ π θ(x)
h
i
R∞
C(y ) = 0 dx PW (x) PW (y − x) = π4 y exp − √2π y
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Summary and Conclusions
• Mean number of common sites visited by N noninteracting random
walkers in all dimensins d

d/2
d <2




As t → ∞

2 < d < dc (N) =
N − d2 (N − 1)
ν=

ν

hCN (t)i ∼ t



0
d > dc (N)
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
2N
N−1
Summary and Conclusions
• Mean number of common sites visited by N noninteracting random
walkers in all dimensins d

d/2
d <2




As t → ∞

2 < d < dc (N) =
N − d2 (N − 1)
ν=

ν

hCN (t)i ∼ t



0
d > dc (N)
2N
N−1
• Full prob. dist. of the number of distinct sites DN (t) and common sites
CN (t) in d = 1 for all N
Exact scaling functions for large N: D(y ) = 2 e −y K0h 2 e −y /2
i
C(y ) = π4 y exp − √2π y
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Summary and Conclusions
• Mean number of common sites visited by N noninteracting random
walkers in all dimensins d

d/2
d <2




As t → ∞

2 < d < dc (N) =
N − d2 (N − 1)
ν=

ν

hCN (t)i ∼ t



0
d > dc (N)
2N
N−1
• Full prob. dist. of the number of distinct sites DN (t) and common sites
CN (t) in d = 1 for all N
Exact scaling functions for large N: D(y ) = 2 e −y K0h 2 e −y /2
i
C(y ) = π4 y exp − √2π y
Open Questions:
• Distributions of DN (t) and CN (t) in higher dimensions d > 1
• Interacting walkers: e.g. Vicious walkers
S.N. Majumdar
Distinct and Common Sites visited by N random Walkers
Growth of the mean no. of distinct sites visited with
time
D N (t)
vs.
I
0
t
III
II
000
111
000
111
111
000
000
111
00
11
00
11
11
00
00
11
t*1
t2*
t
hDN (t)i
∼ td
t1∗ ∼ ln N
t < t1∗
h id/2
∼ t d/2 ln N hD1 (t)i t −d/2
t1∗ < t < t2∗
∼ N hD1 (t)i
t > t2∗
S.N. Majumdar
for all d
t2∗ ∼ ∞
∼ eN
∼ N 2/(d−2)
d <2
d =2
d >2
Distinct and Common Sites visited by N random Walkers