Asymptotics for interacting particle systems onZ d

Bramson, Maury; Griffeath, David

doi:10.1007/bf01013315

Cited by 146 publications

(140 citation statements)

References 9 publications

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“…In the homogeneous source-free case equation (21) reduces toċ = −Dc 2 / ln(1/c), which leads to c(t) ∼ (Dt) −1 ln(Dt) asymptotic behaviour that was rigorously proved to be correct [15].…”

Section: Interacting Particle Systems With a Localized Sourcementioning

confidence: 99%

Probing non-integer dimensions

Krapivsky¹,

Redner²

2007

J. Phys.: Condens. Matter

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Section: Interacting Particle Systems With a Localized Sourcementioning

confidence: 99%

Probing non-integer dimensions

Krapivsky¹,

Redner²

2007

J. Phys.: Condens. Matter

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“…Significantly, any mechanism of stem cell self-renewal, however complex, will lead to clonal evolution that is indistinguishable from one of the three fate behaviours shown in Fig. 3 (Bramson and Griffeath, 1980). Although invariant asymmetry allows individual stem cells to persist long term, both mechanisms of population asymmetry lead to neutral competition between stem cells, as seen in spermatogenesis.…”

Section: Box 1 Neutral Competition Between Stem Cellsmentioning

confidence: 99%

“…In tubular and glandular tissues, clones may expand only along one dimension; in epithelial tissues, clones may expand in two dimensions, and so on. With a stem cell loss rate, l, the average size of persisting clones asymptotes to (Bramson and Griffeath, 1980;Sawyer, 1979;Sudbury, 1976), corresponding to 'tubular' (1D) tissues, 'epithelial' (2D) tissues, 'volumnar' tissues (3D) and 'distributed' (>3D) tissues. Moreover, the asymptotic clone size distributions also vary with dimension F(x)exp [-x 2 /4] in 1D and, as with the cell autonomous process, (15) reveals that daughter cells acquire a symmetric fate every few cell divisions (Clayton et al, 2007).…”

Section: Cell-extrinsic Self-renewalmentioning

confidence: 99%

Universal patterns of stem cell fate in cycling adult tissues

2011

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SummaryIn cycling tissues that exhibit high turnover, tissue maintenance and repair are coordinated by stem cells. But, how frequently stem cells are replaced following differentiation, aging or injury remains unclear. By drawing together the results of recent lineage-tracing studies, we propose that tissue stem cells are routinely lost and replaced in a stochastic manner. We show that stem cell replacement leads to neutral competition between clones, resulting in two characteristic and recurring patterns of clone fate dynamics, which provide a unifying framework for interpreting clone fate data and for measuring rates of stem cell loss and replacement in vivo. Thus, we challenge the concept of the stem cell as an immortal, slowcycling, asymmetrically dividing cell.

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“…The survival probability asymptotics are known in all dimensions, with (1.13) holding in dimensions d ≥ 3 [6]. See [45] for a general introduction to particle systems including the voter model, and see [5], [10], [11] for convergence of rescaled voter models to superBrownian motion.…”

Section: Condition 12 (Cluster Tail Bound)mentioning

confidence: 99%

The survival probability and r-point functions in high dimensions

Hofstad

Holmes

2013

Ann. Math.

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In this paper we investigate the survival probability, θn, in high-dimensional statistical physical models, where θn denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satisfied, then nθn → 2/(AV ), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4 + 1 dimensions, and the spread-out contact process above 4 + 1 dimensions. In the case of oriented percolation, this reproves a result by the first author, den Hollander, and Slade (which was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.

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Asymptotics for interacting particle systems onZ d

Cited by 146 publications

References 9 publications

Probing non-integer dimensions

Probing non-integer dimensions

Universal patterns of stem cell fate in cycling adult tissues

The survival probability and r-point functions in high dimensions

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