ABSTRACT (original article): Self-interacting random walks (SIRWs) show long-range memory effects that result from the interaction of the random walker at time
t
with the territory already visited at earlier times
t'<t
. This class of non-Markovian random walks has applications in contexts as diverse as foraging theory, the behaviour of living cells, and even machine learning. Despite this importance and numerous theoretical efforts, the propagator, which is the distribution of the walker’s position and arguably the most fundamental quantity to characterize the process, has so far remained out of reach for all but a single class of SIRW. Here we fill this gap and provide an exact and explicit expression for the propagator of two important classes of SIRWs, namely, the once-reinforced random walk and the polynomially self-repelling walk. These results give access to key observables, such as the diffusion coefficient, which so far had not been determined. We also uncover an inherently non-Markovian mechanism that tends to drive the walker away from its starting point. CITATION (original article): Julien Brémont, Olivier Bénichou, Raphaël Voituriez (2024), Exact propagators of one-dimensional self-interacting random walks, arXiv:2404.15853. https://doi.org/10.48550/arXiv.2404.15853

π quantities

Here we prove the exactness of the functional equation approach.

Exact expression of the π quantities in the symmetric case

In[]:=
Clear[thcoef,numcoef];​​Γ=3;​​u=
1-Γ
1+Γ
;​​L=30;​​fpp[k_]:=(1+u)(Sinh[(k+1)μ]+
2
u
Sinh[(k-3)μ]+2uSinh[(k-1)μ])((Sinh[kμ]+uSinh[(k-2)μ]));​​fpm[k_]:=(1+u)(Sinh[(k+1)μ]+
2
u
Sinh[(k-3)μ]+2uSinh[(k-1)μ])((1-u)Sinh[μ])​​piexact[k_]:=NProduct[fpp[j],{j,1,L-k}]Product
Sinh[(k-j)μ]
Sinh[(k+1-j)μ]
fpp[L+1-j]+
Sinh[(L-k)μ]
Sinh[(k+1-j)μ]
fpm[L+1-j],{j,1,k};​​piexact[k_]:=NProduct[fpp[j],{j,1,L-k}]Product
Sinh[lμ]
Sinh[(l+1)μ]
fpp[l+(L-k)+1]+
Sinh[(L-k)μ]
Sinh[(l+1)μ]
fpm[l+(L-k)+1],{l,0,k-1};​​μ=
20
L
;​​num=firstindex[L]//Simplify;​​numcoef=Table[SeriesCoefficient[num,{x,0,k}],{k,1,L-1}];​​thcoef=Table[pi[k],{k,1,L-1}];​​funceq=Table[piexact[k],{k,1,L-1}];​​plot=ListPlot[{numcoef,thcoef,funceq},PlotLegends->{"Numerical","Theory (diffusive limit)","Functional eq"},AxesLabel->{"k","π_(k,l)"},PlotMarkers->"OpenMarkers",PlotRange->All]​​(*Export[StringTemplate["Nextcloud/These/Data/satw/Figures/exact-pik-L``-gamma-``-μ-``.pdf"][L,Γ,μ],plot];*)

Exactcomputationofthe
π
k,L-k
atμ=0forSATW

In[]:=
id=IdentityMatrix[2];​​T={{0,1},{1,0}};​​X={{x,0},{0,1}};​​G[k_]:=IdentityMatrix[2]+
Γ
-1+k+Γ
T;​​glist[L_]:=Table[G[k].X,{k,1,L}];​​M[L_]:=Tr[Apply[Dot,Dot@@@Partition[glist[L],UpTo[10]]]]//Simplify;​​fullm[L_]:=Apply[Dot,Dot@@@Partition[glist[L],UpTo[10]]]//Simplify​​fullm[3]+
31
O[x]
//MatrixForm​​fullm[7]+
31
O[x]
//MatrixForm
Out[]//MatrixForm=
Γx
2+Γ
+
2Γ
2
x
2+Γ
+
3
x
+
31
O[x]
1+
2Γx
2+Γ
+
Γ
2
x
2+Γ
+
31
O[x]
Γx
2+Γ
+
2Γ
2
x
2+Γ
+
3
x
+
31
O[x]
1+
2Γx
2+Γ
+
Γ
2
x
2+Γ
+
31
O[x]
Click on the matrix above for higher resolution.
Notice how the ratio of one power k of x on one row from the first column to the second column is ALWAYS equal to
(k)
L-k
. This is the key.

Propagator (computing the double sinh integral)

Computations and contiguous relations

Below are the relevant hypergeometric functions entering in the calculations
In[]:=
fp:=Hypergeometric2F11+ϕ,
1-ϕ
2
,
3(ϕ+1)
2
,
-2β
E
;​​fbmfcm:=Hypergeometric2F1ϕ,
-1-ϕ
2
,
3ϕ+1
2
,
-2β
E
;​​f:=Hypergeometric2F1ϕ,
1-ϕ
2
,
3(ϕ+1)
2
,
-2β
E
;
The formulae below are exact and have been checked. “contig” refers to the fact that we used a contiguous hypergeometric relation to simplify the expressions.
In[]:=
ψ1:=Beta
1+ϕ
2
,ϕ
-βϕ
E
2
ϕ+1
Sinh[β]
fbmfcm;​​ψ2:=Beta
1+ϕ
2
,ϕ
1+ϕ
1+3ϕ
-βϕ
E
2
ϕ+1
Sinh[β]
f;​​ψ1contig:=Beta
1+ϕ
2
,ϕ
-βϕ
E
2
ϕ+1
Sinh[β]
f+fϕ+4
-β

fpϕSinh[β]
1+3ϕ
​​ψ2contig:=Beta
1+ϕ
2
,ϕ
1+ϕ
1+3ϕ
-βϕ
E
2
ϕ+1
Sinh[β]
fbmfcm+2(-1+
-2β

)fpϕ+3fbmfcmϕ
1+ϕ
;​​η1:=Beta
1+ϕ
2
,ϕ
2ϕ
1+3ϕ
-β(ϕ+1)
E
ϕ
Sinh[β]
fp;​​η1contig:=Beta
1+ϕ
2
,ϕ
2ϕ
1+3ϕ
-β(ϕ+1)
E
ϕ
Sinh[β]
1
4ϕ
β

Csch[β](1+3ϕ)Hypergeometric2F1
1
2
(-1-ϕ),ϕ,
1
2
(1+3ϕ),
-2β

-(1+ϕ)Hypergeometric2F1
1-ϕ
2
,ϕ,
3(1+ϕ)
2
,
-2β

;​​η:=
ϕ
Sinh[β]
(η1(2Cosh[u]-Coth[β]Sinh[u])+Sinh[u](ψ1+ψ2));​​ηcontig1:=
ϕ
Sinh[β]
(η1contig(2Cosh[u]-Coth[β]Sinh[u])+Sinh[u](ψ1+ψ2));​​ηcontig2:=
ϕ
Sinh[β]
(η1(2Cosh[u]-Coth[β]Sinh[u])+Sinh[u](ψ1+ψ2contig));​​ηcontig3:=
ϕ
Sinh[β]
(η1(2Cosh[u]-Coth[β]Sinh[u])+Sinh[u](ψ1contig+ψ2));
In[]:=
Clear[ϕ];​​(*FullSimplify[ηcontig1]*)​​FullSimplify[ηcontig2]​​(*FullSimplify[ηcontig3]*)
Out[]=
1
1+3ϕ
-β(3+2ϕ)

Beta
1+ϕ
2
,ϕCsch[β]
β(3+ϕ)

(1+3ϕ)Hypergeometric2F1
1
2
(-1-ϕ),ϕ,
1
2
(1+3ϕ),
-2β

Sinh[u]-2
β(2+ϕ)

ϕHypergeometric2F1
1-ϕ
2
,1+ϕ,
3(1+ϕ)
2
,
-2β


β

Sinh[u]-2Cosh[u]Sinh[β]
In[]:=
FullSimplify[ηcontig1]​​FullSimplify[ηcontig2]​​FullSimplify[ηcontig3]
Out[]=
1
2(1+3ϕ)
-βϕ

Beta
1+ϕ
2
,ϕCsch[β](1+3ϕ)Hypergeometric2F1
1
2
(-1-ϕ),ϕ,
1
2
(1+3ϕ),
-2β

(2Cosh[u]+Sinh[u]-Coth[β]Sinh[u])-(1+ϕ)Hypergeometric2F1
1-ϕ
2
,ϕ,
3(1+ϕ)
2
,
-2β

(2Cosh[u]-(1+Coth[β])Sinh[u])
Out[]=
-β(1+ϕ)

Gamma[ϕ]Gamma
1+ϕ
2
(1+Coth[β])Hypergeometric2F1Regularized
1
2
(-1-ϕ),ϕ,
1
2
(1+3ϕ),
-2β

Sinh[u]+ϕHypergeometric2F1Regularized
1-ϕ
2
,1+ϕ,
3(1+ϕ)
2
,
-2β

(2Cosh[u]-(1+Coth[β])Sinh[u])
Out[]=
1
1+3ϕ
-β(1+ϕ)

Beta
1+ϕ
2
,ϕ
β

(1+ϕ)Csch[β]Hypergeometric2F1
1-ϕ
2
,ϕ,
3(1+ϕ)
2
,
-2β

Sinh[u]+2ϕHypergeometric2F1
1-ϕ
2
,1+ϕ,
3(1+ϕ)
2
,
-2β

(2Cosh[u]+Sinh[u]-Coth[β]Sinh[u])
In[]:=
Clear[ϕ];​​ϕ=4;​​Integrate
ϕ
Sinh[α]
2ϕ+1
Sinh[α+β]
Sinh[α+u]
Sinh[α]
,{α,0,∞},Assumptions->{u>0,β>0,ϕ>0}​​Integrate
ϕ
Sinh[α]
2ϕ+1
Sinh[α+β]
Sinh[β-u]
Sinh[β]
,{α,0,∞},Assumptions->{u>0,β>0,ϕ>0}​​ηfromint=(%%+%)
ϕ
Sinh[β]
//FullSimplify
Check all relations

Contiguous relations

Check that indeed this relation holds
The following is thus the value of 2F1(a+1,b,c,z)
To replace f:
To replace fbmfcm:
Below is valid only for odd ϕ, because it is simpler to check.

Propagator in real space

Position of the bump (taking only two of the terms in the sum)

Diffusion coefficient

Asymmetric case

Computations

Then for η1
The weird Pochhammer formula that makes a lot of things simpler

Plots

Non-monotonicity

Numerically : when are there bumps ?

Regularity

Discontinuities

Derivative

CITE THIS NOTEBOOK

Exact propagators of one-dimensional self-interacting random walks​
by Julien Brémont, Olivier Bénichou and Raphaël Voituriez
Wolfram Community, STAFF PICKS, May 8, 2024
​https://community.wolfram.com/groups/-/m/t/3172230