Prudent walk in dimension six and higher Markus HEYDENREICHLorenzo TAGGINiccol o TORRI Abstract

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Prudent walk in dimension six and higher

Markus HEYDENREICH ∗Lorenzo TAGGI †Niccol`o TORRI ‡

Abstract

We study the high-dimensional uniform prudent self-avoiding walk, which assigns equal prob-

ability to all nearest-neighbor self-avoiding paths of a ﬁxed length that respect the prudent condi-

tion, namely, the path cannot take any step in the direction of a previously visited site. We prove

that the prudent self-avoiding walk converges to Brownian motion under diﬀusive scaling if the

dimension is large enough. The same result is true for weakly prudent walk in dimension d > 5.

A challenging property of the high-dimensional prudent walk is the presence of an inﬁnite-

range self-avoidance constraint. Interestingly, as a consequence of such a strong self-avoidance

constraint, the upper critical dimension of the prudent walk is ﬁve, and thus greater than for the

classical self-avoiding walk.

Keywords: Prudent walk ·Self-avoiding random walk ·Lace Expansion ·Scaling limit ·

Critical dimension

Mathematics Subject Classiﬁcation: 82B41 ·60G50

1 Introduction

Prudent walk is a class of self-repellent random walks where the walk cannot take increments pointing

in the direction of its range. This results in an inﬁnite-range repellence condition. The prudent walk

was originally introduced in [26,27] under the name of self-directed walk and in [21] under the name

outwardly directed self-avoiding walk as a class of self-avoiding walks which are simple to modelize.

In the last 20 years this walk has attracted the attention of the combinatorics community, see e.g.

[1,5,9], and also of the probability community, see e.g. [2,7,18,19].

Let us stress that the prudent condition can be deﬁned in two diﬀerent ways: in its original

formulation [26,27], prudent random walk is a random walk that chooses its direction uniformly

among the admissable moves, and is thus a stochastic process. This model is called kinetic prudent

walk, and was considered in [2]. Alternatively, we ﬁx a length nand choose uniformly a prudent

trajectory of length n, we call this the uniform prudent walk. This latter model has been considered

by the combinatorics community and also investigated probabilistically [18,19]. In this article we

consider the uniform prudent walk.

In dimension d= 2 the scaling limit of the prudent random walk was identiﬁed in [2,18].

The present work concentrates on the high-dimensional case. Similar to other walks without self-

intersections (most notably self-avoiding walk) there exists an upper critical dimension dcsuch that in

dimension d>dcprudent walk is macroscopically very similar to simple random walk. This implies,

in particular, that in high dimensions the various self-repellent walk models are all very similar. Most

interestingly, we ﬁnd strong evidence that this upper critical dimension for prudent walk is dc= 5,

∗Universit¨at Augsburg, Institut f¨ur Mathematik, 86135 Augsburg, Germany. Email:

markus.heydenreich@uni-a.de, Orcid number : 0000-0002-3749-7431

†Sapienza Universit`a di Roma, Dipartimento di Matematica. Piazzale Aldo Moro 5, 00186, Roma, Italy. Email:

lorenzo.taggi@uniroma1.it, Orcid number : 0000-0002-7085-9764

‡MODAL’X, UMR 9023, UPL, Univ. Paris Nanterre, F92000 Nanterre France. Email: ntorri@parisnanterre.fr,

Orcid number : 0000-0002-4778-1305

arXiv:2210.03174v2 [math.PR] 7 Apr 2023

Figure 1: An example of prudent walk path starting at x. When traversing the path starting from x, the tip of the

path at any given moment does not point to the range of the path until that time.

rather than 4 as for self-avoiding walk; the extra dimension results from the inﬁnite-range repellence

condition of prudent walks (see Section 2.6 for further discussion of the upper critical dimension).

Our approach is based on the lace expansion. The lace expansion method was introduced by

Brydges and Spencer [6] to study weakly-self avoiding random walk. The method was widely exploited

for the self-avoiding walk, percolation, lattice trees and lattice animals, and the Ising model, see

[14,16,20,24] and references therein. The present work is the ﬁrst one where we employ the lace

expansion to a model with inﬁnite range interaction.

The lace expansion has also been exploited successfully to investigate critical percolation in high

dimension. It might very well be that the methods proposed in this work to allow for an investigation

of percolation with inﬁnite-range interactions as studied by Hilario and Sidoravicius [15].

1.1 Main results

We let Wnbe the set of n-step nearest-neighbor paths on the hypercubic lattice Zdstarting at the

origin, see (2.3). We call a path w= (w(0), w(1), . . . , w(n)) prudent if w(s)6∈ w(t) + N0w(t)−w(t−

1)for all 0 ≤s<t≤n; see Figure 1for an example.

Our main result is convergence of the rescaled prudent walk to Brownian motion in high dimen-

sion. To this end, let D(A, B) be the space of functions f:A→Bthat are left continuous and have

limits from the right, equipped with the Skorokhod J1-topology, see [3].

For a prudent walk w, we denote the space-time rescaled variable

Xn(t) := 1

√Knwbntc, t ∈[0,1],(1.1)

for a certain constant K > 0 deﬁned in (5.22) below. We consider Xnas a D([0,1],R)-valued random

variable with respect to the uniform measure h·inon the set of n-step prudent walks.

Theorem 1.1 (Convergence to Brownian motion).There exists d0>5such that for d>d0the

following convergence holds. For any bounded continuous function f:D([0,1],Rd)→R, we have that

lim

n→∞ f(Xn)n=Ehf((Ws)s∈[0,1])i

where (Ws)s≥0denotes the standard Brownian motion and Eis its expectation.

The result remains true for the weakly prudent walk introduced in Section 2.3 in dimension d > 5

provided that a “strength parameter” λfor the weakly prudent walk, deﬁned in (2.15), is suﬃciently

small.

We furthermore prove that the prudent bubble condition is satisﬁed if the dimension is large (or

d > 5 and λis small), see Section 2.4.

1.2 Organisation of the paper

We deﬁne prudent walk and weakly prudent walk in Section 2. Our results are stated in Section

2.4, followed by a discussion in Section 2.5. In Section 3we derive the lace expansion for prudent

walks and obtain diagrammatic estimates on the expansion coeﬃcients. Section 4establishes the

convergence of the expansion and establishes the prudent bubble condition. The analysis presented

in these two sections is the major novelty of the present work. Finally, in Sections 5and 6, we prove

the convergence of the prudent walk to a Brownian motion.

2 Deﬁnitions and results

Given a square-sumable function fon Zd, we deﬁne its Fourier transform as

f(k) = X

x∈Zd

f(x)eix·k, k ∈[−π, π]d(2.1)

and observe that

f(x) = Z[−π,π]d

ddk

(2π)db

f(k)e−ik·x.(2.2)

We now recall some basic deﬁnitions for the simple random walk and introduce the main deﬁnitions

for prudent walk and weakly prudent walk.

2.1 Preliminaries on simple random walk

For x, y ∈Zdand n∈N0:= {0,1,2, . . .}, we write

Wn(x, y) := nw:{0, . . . , n} → Zd:w(0) = x, w(n) = y,

and |w(s)−w(s−1)|= 1 for all s= 1, . . . , no(2.3)

for the set of n-step walks from xto y. We further write W(x, y) = Sn∈N0Wn(x, y), and for w∈

W(x, y) we write |w|for the length of the walk, that is, the unique n∈N0such that w∈ Wn(x, y).

We denote by Czthe simple random walk Green’s function, that is,

Cz(x, y) := ∞

n=0 Wn(x, y)zn,(2.4)

and we let Cz(x) = Cz(0, x). We recall that the radius of convergence of z7→ Px∈ZdCz(x) equals

2d. It is convenient to introduce the function

D(x) := (1

2dif |x|= 1,

0 otherwise,x∈Zd,(2.5)

with Fourier transform given by b

D(k) = 1

dPd

i=1 cos(ki). Since |Wn(0, x)|= (2d)nD∗n(x) (where

D∗n=D∗···∗Ddenotes n-fold convolution), we get that the Fourier transform of Cz(x) for |z|< zc

is given by

Cz(k) = 1

1−2dz b

D(k), k ∈[−π, π]d.(2.6)

For later reference, we note the elementary relations

1−b

D(k) = 1

2d+o(1)|k|2as |k| → 0and lim

n→∞ n1−b

D(k/√n)=1

2d|k|2,(2.7)

where |·|denotes the L2norm.

2.2 The prudent walk

We now properly deﬁne the prudent walk.

Deﬁnition 2.1. A walk w∈ Wn(x, y)satisﬁes the prudent condition if

w(s)6∈ w(t) + N0w(t)−w(t−1)for all 0≤s<t≤n.(2.8)

We shortly say “wis prudent”. The set of prudent n-step walks from xto yis denoted by Pn(x, y)

and cn(x, y) := |Pn(x, y)|denotes its cardinality. We denote cn(x) := cn(0, x). The total number of

n-step prudent walks is cn:= Px∈Zdcn(x). From sub-additivity it follows that the limit

µ:= lim

n→+∞(cn)1/n (2.9)

exists. The constant µis called the connective constant and satisﬁes the trivial bounds

d−1≤µ≤2d−1.(2.10)

Prudent random walk is the uniform measure on the set of n-step prudent walks. Note that the

prudent random walk is not a stochastic process, because the resulting family of measures is not

consistent.

The prudent two-point function is deﬁned as the generating function

Gz(x, y) = ∞

n=0

cn(x, y)zn(2.11)

and we let Gz(x) = Gz(0, x). We let zc:= 1

µbe the radius of convergence of the power series

χ(z) := ∞

n=0

cnzn=X

x∈Zd

Gz(x).(2.12)

We refer to zcas critical point and χ(z) as susceptibility.

2.3 The weakly-prudent walk

We now introduce the weakly-prudent walk, in which each step of the walk which does not fulﬁll

the prudent condition is penalized by a multiplicative parameter λ∈[0,1]. Hence the case λ= 0

corresponds to simple random walk, while the case λ= 1 corresponds to prudent walk. We also

rephrase the expression for the prudent two-point function in (2.11) so that it is more suited for an

expansion. Given a n-step walk w= (w(0), . . . , w(n)), and 0 ≤s<t≤n, we deﬁne

Ust(w) := (−1,if w:w(t)↓w(s),

0,otherwise,(2.13)

where w(t)↓ais shorthand for a−w(t)∈N0w(t)−w(t−1)—recall the prudent condition (2.8).

If wis such that w(t)↓a, then we say that the t-th step of wsees aor, shortly, w(t)sees a. The

two-point function for the weakly-prudent walk is deﬁned as

Gλ

z(x, y) = X

n≥0

zncλ

n(x, y) (2.14)

where

cλ

n(x, y) = X

w∈Wn(x,y)

ϕλ(w) and ϕλ(w) := Y

0≤s<t≤n1 + λ Ust(w).(2.15)

We also let cλ

n(x) = cλ

n(0, x) and Gλ

z(x) = Gλ

z(0, x).

We let cλ

n=Px∈Zdcλ

n(x). Let us observe that ϕ1(w)6= 0 if and only if wsatisﬁes the prudent

condition. To simplify the notation, we denote ϕ(w) = ϕ1(w).

In analogy with (2.12), for any λ > 0, we deﬁne zc(λ) as the radius of convergence of the

susceptibility

χλ(z) := X

x∈Zd

Gλ

z(x) = X

n≥0

znX

w∈Wn

ϕλ(w).(2.16)

Hence it follows that zc(0) = 1

2d,zc(1) = 1

µ,G1

z(x) = Gz(x),and χ1(z) = χ(z).

2.4 Results

A central object in our analysis is the prudent bubble diagram, which we now introduce. To this end,

we introduce a notion that is weaker than the symbol w(t)↓aused in (2.13): for x, a ∈Zd,a6=x,

we let

x⊥a⇐⇒ x−a∈[

j=1,...,d

ejZ,(2.17)

where e1, . . . , edare the coordinate axes (i.e., x⊥awhenever xand adiﬀer in at most one coordinate).

For any x, a ∈Zdwe deﬁne the modiﬁed indicator function

1{⊥a}(x) := (1

dif x⊥aand x6=a

0 otherwise,(2.18)

and abbreviate 1⊥(x) = 1{⊥0}(x). Observe that 1{⊥a}(x) = 1⊥(x−a).

Deﬁnition 2.2 (Prudent Bubble Diagram).We deﬁne the prudent bubble diagram as

Bλ

z:= kGλ

z∗Gλ

z∗1⊥k∞= sup

y∈ZdX

x1,x2∈Zd

Gλ

z(0, x1)Gλ

z(x1, x2)1{⊥y}(x2) (2.19)

If λ= 1, we write Bz=B1

The next theorem states that the critical prudent bubble diagram is ﬁnite if the dimension is

suﬃciently large and λ= 1 or if the dimension is at least 6 and λis suﬃciently small.

Theorem 2.1 (Prudent Bubble Condition).There exist d0≥5and λ0>0such that if either

(a) d > 5and λ∈(0, λ0), or

(b) λ= 1 and d > d0,

then the critical prudent bubble diagram Bλ

zcis ﬁnite. Moreover, there exists a constant C > 0

(independent of λand the dimension d) such that

Bλ

zc< C/d. (2.20)

The proof of Theorem 2.1 is intertwined with the asymptotic behaviour of the critical two-point

function, which we formulate in the next theorem.

Theorem 2.2 (Critical Two-Point Function).Under the conditions of Theorem 2.1, there exists

K∈(0,∞), deﬁned in (5.22) below, such that for k∈Rd,

bcλ

nk/√n∼bcλ

n0exp{−K|k|2}as n→ ∞.(2.21)

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PrudentwalkindimensionsixandhigherMarkusHEYDENREICH*LorenzoTAGGINiccoloTORRIAbstractWestudythehigh-dimensionaluniformprudentself-avoidingwalk,whichassignsequalprob-abilitytoallnearest-neighborself-avoidingpathsofaxedlengththatrespecttheprudentcondi-tion,namely,thepathcannottakeanystepinthedirect...

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