Occupation-condensation transition of a sublinearly vertex-reinforced random walk on regular tree

Bon A Koo; Edward Ju

REVIEW 4 major objections 5 minor 11 references

Occupation-condensation transition of a sublinearly vertex-reinforced random walk on regular tree

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A sublinearly reinforced random walk on a b-ary tree condenses its occupation onto one vertex above a finite strength β_c(a,b), with the range still growing slowly — a sharp transition stable out to t=3×10^7.

desk verdict Genuinely new numerics for sublinear VRRW on trees, honest hedging, but the sharp-transition claim is finite-time and the paper misstates what is known on Z (a>1/2 was settled by Basdevant–Schapira–Singh). read the letter →

arxiv 2607.16971 v1 pith:6RS5JABC submitted 2026-07-18 cond-mat.stat-mech math-phmath.MP

Bon A Koo , Edward Ju This is my paper · ORCID

classification cond-mat.stat-mechmath-phmath.MP MSC 60K3582C4160J10

keywords vertex-reinforcedrandomwalkcondensationtransitionoccupationmeasureregulartreebranchingnumberreversibleMarkovchainrangeexponentsublinearreinforcement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the sublinear vertex-reinforced random walk on a regular tree — a walk that favors revisiting neighbors in proportion to 1+β n^a — has a genuine condensation transition in the occupation measure. Below a finite critical reinforcement β_c(a,b), the walk spreads: the most-visited vertex holds a vanishing fraction of time and the range grows linearly. Above β_c, a single vertex captures an order-one fraction of the walk's time, stable over the observation window, while the walk continues to discover new vertices at a very slow, effectively logarithmic, rate. The authors argue this is occupation condensation, not finite-range localization, because the support keeps growing. If true, it provides a computable example of a self-built environment condensing a random walk's occupation, and it separates the role of the geometry (branching number b) from the memory exponent a.

What carries the argument

The argument runs on three linked objects. (1) The frozen-environment reversibility: for fixed weights w_v=1+β n_v^a, the walk is reversible with stationary measure μ_v ∝ w_v W_v, W_v=∑_{u∼v} w_u, and edge conductances c_uv=w_u w_v. (2) The core recursion: on the backbone, μ_v ∝ w_v W_v reduces to the nonlinear three-term recursion μ(d)=C μ(d)^a [μ(d−1)^a + μ(d+1)^a], which fixes the shape of the condensed droplet and identifies a=1/2 as the marginal exponent for geometric tails. (3) The frontier balance: reading the frozen walk via the branching-number criterion for biased walks on trees — escape while the backtrack weight λ_eff=1+β n_p^a is below b — gives n* = ((b−1)/β)^{1/a} and, through

What would settle it

Run the same walk at a=0.6, b=2 to t ≥ 10^8 with hundreds of realizations and check whether the ν_R=1/2 crossing keeps moving; or measure the waiting-time distribution between discoveries of new vertices in the condensed phase — if the hazard of discovering a new vertex decays so that R(t) saturates, the 'slowly unbounded range' picture would fail; alternatively, if m1 at β>β_c decays toward zero at longer times, there is no stable condensate.

Watch

Extended reading notes

Core claim

The central discovery is a sharp condensation transition for the occupation measure of the sublinear VRRW on the rooted b-ary tree. At β below β_c(a,b), the range exponent ν_R tends to 1 (linear spreading) and m1, the maximum visit fraction, tends to zero. Above β_c, m1 tends to an O(1) constant — e.g. 0.27 at β=2.5, t=10^5, and 0.28 at t=10^7 — while the range R(t) keeps growing, with local exponent ν_R ≈ 0.1 and held-out fits favoring log t over power laws; R is not found to be bounded. Four estimators (m1 half-height, susceptibility peak, Binder crossing, and ν_R=1/2) locate the same threshold, with the ν_R=1/2 crossing showing no systematic drift for t up to 3×10^7. In the condensed phas

Load-bearing premise

The central claim of a sharp, stable transition rests on finite-time simulations with 24 realizations up to t=3×10^7; the crossing-point scatter of order 0.1 bounds but does not exclude a slow drift, and the distinction between logarithmically growing and slowly power-law-growing range is made on very small counts.

Editorial extensions

If this is right

The occupation measure of the sublinear VRRW on a tree has two distinct long-time regimes separated by a finite β_c: linear spreading and condensate-plus-slow-spreading; the condensate fraction, not the range, is the right order parameter.
The transition line obeys β_c ∝ b−1 over the tested range, so the branch factor sets the threshold while the memory exponent a shapes it; the same law predicts β_c→0 as b→1.
The condensed core is described by the reversible measure μ_v ∝ w_v W_v, allowing quantitative predictions of occupation profiles, including the non-monotone droplet shape peaked a few levels below the root.
The a=1/2 exponent from the one-dimensional problem is not the threshold on trees; it governs only the marginal shape of the core, decoupled from the frontier escape that sets β_c.
Near the transition, run-to-run fluctuations do not vanish with time (non-self-averaging, bimodal m1, negative Binder cumulant), a coexistence-type signature that would carry over to a genuine first-order transition if the finite-time modes stabilize.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the transition is genuine, the condensation-versus-localization distinction should matter on Z^d: the natural order parameter is the condensate fraction, not the range, and sublinear reinforcement may produce a condensate with unbounded support there as well.
The recursion (6) is of the type that generates non-uniform scaling; the occupation measure on the core may be multifractal, and computing its spectrum would be a natural test of the recursion's universality.
The scaling law β_c ∝ b−1 invites a test on non-regular trees: replace b−1 by the branching number br(T) of the tree and check collapse on Galton–Watson or other random trees.
The apparent threshold near a≈0.4 could be either a genuine a_c>0 or a steep divergence of β_c; distinguishing them requires probing larger β or developing a dynamic (non-frozen) treatment, perhaps along the lines of exactly solvable edge-reinforced cases.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague — this one is worth engaging, but keep the abstract's 'sharp transition' in quotes when you summarize it. The paper is honest: it repeatedly flags what is ansatz and what is open, it does not claim a bounded range or a proven first-order transition, and it ships code and data (on acceptance). The genuinely new pieces: sublinear VRRW on b-ary trees was open; the reversible-core description µ ∝ w_v W_v is tested by the Q(d) estimator, which is non-circular and flat across the core; and the β_c ∝ b−1 collapse in Table 1 holds with bootstrap CIs across b=2,3,4 at several values of a. That collapse is an empirical scaling law — the closure through \bar m(a,b) is read backward from the data, so the 'parameter-free prediction' label overstates the evidence, but the collapse itself is real. The a=1/2 marginal-profile statement from the recursion is exact and clean.

The soft spots are where you'd expect them. The sharpness of the transition rests on 24 realizations out to 3×10^7: the ν_R=1/2 crossings (1.56, 1.69, 1.50, 1.55) bound but do not exclude a slow drift in β_c, and the authors themselves concede that the coexistence signatures could be a broad distribution of trapping times. That is acceptable for a numerical paper, but the title and abstract overstate the asymptotics. The four-estimator agreement is at one parameter point; Figure 1 uses 8 realizations without error bars; and the slow-phase range counts (19 to 48) make the log-vs-power comparison suggestive, not decisive.

One thing that will annoy the subfield: the introduction and Remark 1 say Chen–Kozma left a∈[1/2,1) open on Z. That is wrong — Basdevant, Schapira and Singh, cited as [9] but never mentioned in the text, proved localization for a>1/2. The paper's own condensation-vs-localization distinction is exactly their territory and needs to be engaged.

Verdict: send it to a serious referee. Reproducible numerics, a non-circular core test, and a clean citable scaling law. Fix the literature error and soften the sharp-transition language and it is a solid contribution to the self-interacting random walk literature.

Referee Report

4 major / 5 minor

Summary. The paper studies a vertex-reinforced random walk on the rooted b-ary tree with sublinear reinforcement weights w_v=1+β n_v^a. It claims a condensation transition at finite β_c(a,b): below β_c the range grows linearly and the occupation spreads; above β_c a single vertex carries an O(1) fraction of the time while the range still grows very slowly, better described by log t than by a power. The authors identify the transition via the ν_R=1/2 crossing of the range exponent, confirm it with four independent estimators at one parameter point (a=0.6,b=2), and report no systematic drift out to t=3×10^7. They also derive a reversible stationary measure for the frozen condensed core, μ_v∝w_v∑_{u∼v}w_u, test the neighbour coupling non-circularly through the ratio Q(d), and propose a branching-number law β_c∝b−1, validated by collapse of the measured lines for b=2,3,4.

Significance. If the finite-time extrapolation is correct, this is a valuable contribution to the largely open regime of sublinear vertex reinforcement. The paper supplies a numerical phase diagram, a reversible quasi-stationary measure for the condensed core, a direct test of the neighbour coupling that avoids the obvious self-counting inflation, and a b−1 scaling law with bootstrap confidence intervals. The authors are unusually careful: they distinguish occupation condensation from finite-range localization, state that the frozen-environment argument is an adiabatic scaling ansatz, and explicitly disclaim a proven first-order transition. The promise of reproducible code and scripts is a further strength. The main reservation is that the central 'sharp transition' claim rests on finite-time, small-sample extrapolation; the paper would be stronger if the language were matched to what the data actually establish.

major comments (4)

[Section 6, Fig. 4] The β_c(t_max) sequence 1.56, 1.69, 1.50, 1.55 over t_max=10^5 to 3×10^7, with only 24 realizations and scatter ~0.1, bounds but does not exclude a slow drift of order 0.1 per decade. The abstract's 'sharp transition' and 'no systematic drift' overstate what the data show. The authors' own caveat that the coexistence signatures may reflect a broad distribution of trapping and escape times is the correct reading. Please either add per-point uncertainties and phrase the claim as 'no detectable drift within resolution', or explicitly recast the transition as an effective finite-time crossover whose asymptotic nature is unresolved.
[Section 5, Eq. (11)] The branching-number law β_c=(b−1)/\bar m(a)^a is closed through \bar m(a), which is defined by inverting the measured β_c; it is not measured independently. The b-collapse in Table 1 therefore demonstrates self-consistency of the one-parameter scaling ansatz, but it does not validate the frozen-environment criterion as a predictive mechanism. The abstract's wording 'predicting β_c ∝ b−1' is too strong. Either measure \bar m microscopically from the frontier visit-count distribution, or present \bar m explicitly as an effective fitted parameter and describe the b-collapse as a consistency check.
[Section 3 and Section 6] The entire phase map in Figure 1 is based on only 8 realizations, with no error bars, and the ν_R=1/2 definition is used as a proxy for the condensation transition. The four-estimator agreement that justifies this proxy is demonstrated at a single point, a=0.6, b=2, with no check that the m_1-based markers coincide with ν_R=1/2 elsewhere in the (a,β) plane or for other b. To support the comprehensive phase diagram, at least one additional (a,b) point with the full set of markers, and error estimates for the Figure 1 sweeps, are needed.
[Section 6, range growth] Above β_c the range counts are very small: at β=6, R grows only from 19 at t=4×10^4 to 48 at t=3×10^7. The held-out extrapolation favouring log t over a power law is based on these small counts, so the claim that growth is 'better described by log t than by any power' is not established as an asymptotic statement. Because the distinction between condensation and bounded-range localization depends on whether the range continues to grow, this needs confidence intervals on the extrapolation errors or longer data before it can be used as a central conclusion.

minor comments (5)

[Figure 1] The caption should state the number of realizations (8) and explicitly note that no error bars are shown. The fit interval for ν_R used in this figure is given in Section 2 but should be repeated in the caption.
[Section 6] The statement 'four estimators locate the same threshold' appears in the abstract without qualification; the agreement is demonstrated at a=0.6, b=2. Rephrase to avoid implying the full check was done in the whole phase plane.
[Section 5, Table 1] The bootstrap confidence intervals would be clearer if the resampling scheme were stated explicitly (e.g., resampling over realizations or over time blocks).
[Section 4, Eq. (6)] The normalization constant C and the boundary conditions of the recursion (at the root and at the edge of the core) are not specified. A short explanation would help the reader reproduce the numerical relaxation described in the text.
[Data availability] The repository is 'made public on acceptance'; to substantiate the reproducibility claim at review time, consider providing a reviewer link or stating that code is available on request.

Circularity Check

1 steps flagged · score 4.0 of 10

Branching-number law is a scaling ansatz closed by a fitted scale; central transition and reversible-core findings remain independent.

fitted input called prediction [Section 5, Eq. (11) and Figure 3]
"To place a line in the (a, β) plane we close (10) through a single occupancy scale ¯m(a, b) for the vertex behind the edge, writing the crossover as β¯ma = b−1, so that βc(a, b) = b−1 / ¯m(a)^a, (11), provided ¯m depends only weakly on b. This is a scaling ansatz, not a derivation; its two content-bearing predictions are βc ∝ b−1 at fixed a... The occupancy scale read back from (11), ¯m(a) = ((b−1)/βc)^{1/a} (right panel), is b-independent."

The scale ¯m is not measured independently; it is defined from the measured βc via Eq. (11). Therefore the statement that ¯m is b-independent is exactly equivalent to the statement that βc/(b−1) is b-independent, which is precisely the collapse being tested. The branching-number criterion alone gives only the local drift-balance n*; the b-independent occupancy scale is imposed as part of the ansatz. Thus the claimed prediction 'βc ∝ b−1' is a restatement of the ansatz's defining condition rather than an external consequence of the criterion. The collapse confirms the scaling hypothesis but does not independently test a separate prediction.

full rationale

The paper's central claims—a sharp condensation transition at finite βc, the O(1) condensate with slowly growing range, and the reversible core measure—are supported by direct simulation diagnostics and are not circular. The Q(d) test in Eq. (7) is explicitly constructed to avoid the self-count inflation and is an independent check of the neighbour coupling. The four markers of the transition (m1 half-height, susceptibility peak, Binder crossing, νR=1/2) are independent and agree. The observation-time stability is presented with the caveat that 24 realizations and finite t bound but do not exclude slow drift, and the authors explicitly disclaim a proven first-order transition. There is no load-bearing self-citation: the external results cited (branching-number criterion, Chen–Kozma, Pemantle, Sabot–Tarrès) are rigorous prior work, and the paper's own contribution is not hidden behind them. The only genuine circularity is in the branching-number law: Eq. (11) is closed through the fitted scale ¯m, whose b-independence is mathematically equivalent to the βc ∝ (b−1) collapse being presented as a prediction. This makes that particular 'prediction' a self-consistent restatement of the scaling ansatz, a partial circularity in a secondary organizational claim rather than in the central transition result. Hence the score is 4.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

Most physical content is either exact elementary Markov-chain identities or numerical observations. The only genuinely free quantity is the effective scale mbar in Eq. (11), read backward from the data it is used to explain. The adiabatic use of the frozen-tree transience criterion is a stated assumption, not a derived theorem.

free parameters (1)

mbar(a,b) = Read back from measured β_c as ((b−1)/β_c)^(1/a); reported values rise from near 0 at a≈0.45 to about 1.5 at a=0.75 (Fig
Introduced in Eq. (11) to close the frontier balance. Its b-independence is asserted, not independently measured; reading it backward from the same β_c data makes the β_c ∝ b−1 'prediction' a restatement of the ansatz rather than a derived law.

assumptions (5)

standard math Branching-number criterion: a biased nearest-neighbor walk on the b-ary tree stepping to each child with weight 1 and to the parent with weight λ is transient iff λ < b.
Used in Section 5, Eq. (10), to estimate where the local drift at the frontier reverses. Cited to Lyons and Lyons–Peres; rigorous for a frozen environment.
standard math In a frozen environment, the chain with transition P(v→u)=w_u/W_v is reversible with stationary measure µ_v ∝ w_v W_v and edge conductances c_uv = w_u w_v.
Detailed balance is checked explicitly in Section 4; this is elementary Markov-chain theory.
domain assumption On the condensed core, weights grow as w_v(t) ≃ β t^a µ_v^a for vertices with positive limiting occupation µ_v.
Section 4 uses this to reduce the occupation law to the recursion (6). It is an asymptotic ansatz about how the self-built environment freezes on the core, not proven for the dynamic process.
ad hoc to paper The frozen-environment branching-number criterion applies adiabatically to the dynamically evolving frontier of the reinforced walk.
Section 5 explicitly says the criterion is used as a 'frozen-environment estimate' and that the closure through mbar is a 'scaling ansatz, not a derivation'. This is the load-bearing modeling assumption for the β_c ∝ b−1 law.
ad hoc to paper A single effective occupancy scale mbar(a,b), weakly dependent on b, suffices to close the frontier balance.
Eq. (11) sets β_c = (b−1)/mbar^a. The b-independence of mbar is exactly the content of the β_c ∝ b−1 collapse; it is assumed, then read backward from the fitted β_c.

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Cite this review

Pith. "Pith review of Occupation-condensation transition of a sublinearly vertex-reinforced random walk on regular tree." pith.science (2026). https://pith.science/paper/6RS5JABC

@misc{pith2026260716971,
  author       = {Pith},
  title        = {Pith review of: Occupation-condensation transition of a sublinearly vertex-reinforced random walk on regular tree},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6RS5JABC}},
  note         = {Machine review of arXiv:2607.16971}
}

abstract

A vertex-reinforced random walk steps to a neighbour with probability proportional to $1+\beta n^{a}$, where $n$ counts previous visits to that neighbour and $a\in(0,1)$ sets the memory strength. On the rooted $b$-ary tree the exponential growth of the vertex set drives the walk outward while the reinforcement pulls it back. We report a sharp condensation transition of the occupation measure at a finite $\beta_c(a,b)$: below it the occupation spreads and the range grows linearly; above it a single vertex holds an $O(1)$ fraction of the time, stable in the observation time, while the range keeps growing very slowly, at a rate better described by $\log t$ than by any power. We do not find the range to be bounded, and keep this condensation distinct from finite-range localization. Four estimators locate the same threshold, which shows no systematic drift out to $t=3\times10^{7}$. In a frozen environment the walk is reversible, with edge conductances $c_{uv}=w_{u}w_{v}$, $w_{v}=1+\beta n_{v}^{a}$, and measure $\mu_{v}\propto w_{v}\sum_{u\sim v}w_{u}$ describing the condensed core, whose neighbour coupling we test directly. Reversibility places the escape at the frontier within the branching-number criterion for biased walks on trees, predicting $\beta_c\propto b-1$; the measured lines for $b=2,3,4$ collapse under division by $b-1$ to a few percent (bootstrap). The value $a=1/2$ that governs the walk on $\mathbb{Z}$ enters only as the marginal exponent of the condensed profile. Near $\beta_c$ the occupancy is non-self-averaging and bimodal, a coexistence-type phenomenology.

Figures

Figures reproduced from arXiv: 2607.16971 by the authors.

**Figure 2.** Occupation of the condensed core (a = 0.7, β = 4, t = 3 × 105 ). Left: measured occupation µv against the reversible prediction wvWv of Eq. (4) (both normalized); the dashed line is µv = wvWv. Right: occupation of the most visited vertex at each depth, showing the core droplet peaked a few levels below the root. enters only through the b−1 off-backbone vertices that were dropped, which control the escape at the fron… view at source ↗

**Figure 3.** The branching-number law (11). Left: measured lines [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Observation-time analysis at a = 0.6, b = 2, to t = 3 × 107 (24 realizations). Left: mean range R(t); the range grows for every β, linearly below βc and very slowly above (the ∝ t 0.1 guide is an effective-exponent reference; held-out extrapolation favours logarithmic …

**Figure 5.** Figure 5: Occupancy statistics at a = 0.6, b = 2, with t from 105 to 107 . From left: ensemble mean ⟨m1⟩; Binder cumulant U, which tends to 2/3 in the condensed phase and dips negative on the spreading side; susceptibility χ = t Var(m1), whose peak near β ≈ 1.5 grows with t; and…

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Reference graph

Works this paper leans on

11 extracted references

[9]

Basdevant, B

A.-L. Basdevant, B. Schapira, A. Singh, Localization of a vertex- reinforced random walk onZwith sub-linear weight, Probab. Theory Related Fields 159 (2014) 75–115

2014
[1]

Pemantle, A survey of random processes with reinforcement, Probab

R. Pemantle, A survey of random processes with reinforcement, Probab. Surv. 4 (2007) 1–79

2007
[2]

Davis, Reinforced random walk, Probab

B. Davis, Reinforced random walk, Probab. Theory Related Fields 84 (1990) 203–229

1990
[3]

Benaïm, Vertex-reinforced random walks and a conjecture of Peman- tle, Ann

M. Benaïm, Vertex-reinforced random walks and a conjecture of Peman- tle, Ann. Probab. 25 (1997) 361–392

1997
[4]

Tarrès, Vertex-reinforced random walk onZeventually localizes on five sites, Ann

P. Tarrès, Vertex-reinforced random walk onZeventually localizes on five sites, Ann. Probab. 32 (2004) 2650–2701

2004
[5]

Pemantle, Phase transition in reinforced random walk and RWRE on trees, Ann

R. Pemantle, Phase transition in reinforced random walk and RWRE on trees, Ann. Probab. 16 (1988) 1229–1241

1988
[6]

Sabot, P

C. Sabot, P. Tarrès, Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model, J. Eur. Math. Soc. 17 (2015) 2353–2378

2015
[7]

J. Chen, G. Kozma, Vertex-reinforced random walk onZwith sublinear weight, Proc. Lond. Math. Soc. 109 (2014) 1538–1568. 18

2014

Show all 11 references

[8]

Volkov, Vertex-reinforced random walk on arbitrary graphs, Ann

S. Volkov, Vertex-reinforced random walk on arbitrary graphs, Ann. Probab. 29 (2001) 66–91

2001
[10]

Lyons, Random walks and percolation on trees, Ann

R. Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990) 931–958

1990
[11]

Lyons, Y

R. Lyons, Y. Peres, Probability on Trees and Networks, Cambridge Uni- versity Press, Cambridge, 2016. 19

2016

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [9]

Basdevant, B

A.-L. Basdevant, B. Schapira, A. Singh, Localization of a vertex- reinforced random walk onZwith sub-linear weight, Probab. Theory Related Fields 159 (2014) 75–115

2014

[2] [1]

Pemantle, A survey of random processes with reinforcement, Probab

R. Pemantle, A survey of random processes with reinforcement, Probab. Surv. 4 (2007) 1–79

2007

[3] [2]

Davis, Reinforced random walk, Probab

B. Davis, Reinforced random walk, Probab. Theory Related Fields 84 (1990) 203–229

1990

[4] [3]

Benaïm, Vertex-reinforced random walks and a conjecture of Peman- tle, Ann

M. Benaïm, Vertex-reinforced random walks and a conjecture of Peman- tle, Ann. Probab. 25 (1997) 361–392

1997

[5] [4]

Tarrès, Vertex-reinforced random walk onZeventually localizes on five sites, Ann

P. Tarrès, Vertex-reinforced random walk onZeventually localizes on five sites, Ann. Probab. 32 (2004) 2650–2701

2004

[6] [5]

Pemantle, Phase transition in reinforced random walk and RWRE on trees, Ann

R. Pemantle, Phase transition in reinforced random walk and RWRE on trees, Ann. Probab. 16 (1988) 1229–1241

1988

[7] [6]

Sabot, P

C. Sabot, P. Tarrès, Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model, J. Eur. Math. Soc. 17 (2015) 2353–2378

2015

[8] [7]

J. Chen, G. Kozma, Vertex-reinforced random walk onZwith sublinear weight, Proc. Lond. Math. Soc. 109 (2014) 1538–1568. 18

2014