Pith. sign in

REVIEW 3 major objections 4 minor 43 references

Critical Parameters for Loop and Bernoulli Percolation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Infinite loops arise strictly later than infinite percolation clusters

desk verdict Likely-correct strict inequality for loop vs percolation thresholds on bounded-degree graphs, with a proof that is intricate and needs referee work, but the stress-test's counterexample does not land. read the letter →

arxiv 1908.10213 v1 pith:4BECDABH submitted 2019-08-27 math.PR math-phmath.MP

Peter Mühlbacher This is my paper · ORCID
classification math.PRmath-phmath.MP MSC 60K3582B4360G5582B20
keywords randomloopmodelsinterchangeprocessbondpercolationcriticalparameterstochasticdominationboundeddegreephasetransition
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 proves that on every connected, countably infinite graph with uniformly bounded vertex degree, a natural random loop model (including the random interchange process) develops infinite loops only at a strictly larger parameter $\beta_c(u)$ than the parameter $\beta_c^{\mathrm{per}}$ at which the coupled Bernoulli bond-percolation model develops infinite clusters. Equivalently, there is an interval of $\beta$ values in which infinite percolation clusters exist with positive probability while every loop visits only finitely many vertices almost surely. The result holds for all loop-type parameters $u \in (0,1]$ and gives, on $\mathbb{Z}^d$ for $d\ge2$, exponential decay of loop sizes throughout the gap interval. It settles, for bounded-degree graphs, the question of whether the natural lower bound $\beta_c \ge \beta_c^{\mathrm{per}}$ can be strict, in contrast with graphs of diverging degree where equality is conjectured and known in several mean-field cases.

What carries the argument

The carrying object is the red-edge process $(R_e)$, a local configuration event on each edge. Proposition 3.1 shows that, conditioned on every edge of any subgraph carrying at least one link, the law of $(R_e)_{e\in E'}$ stochastically dominates a product Bernoulli measure with parameter $\delta>0$ depending only on $\beta$ and the degree bound $\Delta$, uniformly over all admissible boundary conditions. The proof of Proposition 3.1 uses a spatial Markov property to reduce arbitrary conditioning to a two-layer neighbourhood of the edge, a classification of pivotal neighbouring edges, and a final reduction to the law of a Poisson point process on a set of intervals of length at least $\beta/2$; this yields a strictly positive uniform lower bound on the conditional probability that a given edge is red.

What would settle it

Fix $u=1$ on $\mathbb{Z}^2$ and simulate the loop model at $\beta=-\ln(1-p_c(\mathbb{Z}^2))$: if infinite loops occur with positive probability at that $\beta$, Theorem 2.1 is false. Alternatively, construct a sequence of admissible boundary conditions on the two-neighbourhood of an edge $e_0$ for which the conditional probability that $e_0$ is red tends to zero; this would contradict Proposition 3.1 and break the proof.

Watch

Extended reading notes

Core claim

Theorem 2.1: for all countably infinite connected graphs $G$ of uniformly bounded degree with $p_c(G,\mathrm{bond})<1$ and all $u\in(0,1]$, the strict inequality $\beta_c(u)>\beta_c^{\mathrm{per}}$ holds, where $\beta_c^{\mathrm{per}}=-\ln(1-p_c)$. The proof couples the loop configuration to bond percolation with $p=1-e^{-\beta}$ and colours each edge red, blue, or uncoloured: an edge is red when it carries exactly two crosses and no neighbouring edge has a link between them; blue when it carries at least one link but is not red. Loops can travel only along blue edges, so if red edges are abundant enough and form an independent-like set, deleting them from a barely supercritical percolation cuts every infinite cluster while the original percolation still percolates. This yields a gap interval on every bounded-degree graph, and in particular on $\mathbb{Z}^d$ an interval with infinite percolation clusters but exponentially small loops.

Load-bearing premise

The proof rests on the claim that conditioned on every edge of a subgraph carrying at least one link, the red-edge indicators stochastically dominate a product Bernoulli measure with a strictly positive parameter $\delta$ that is uniform over all boundary conditions; if that uniform domination failed, deleting red edges from a barely supercritical percolation could leave an infinite cluster of blue edges.

Editorial extensions

If this is right

  • On $\mathbb{Z}^d$, $d\ge2$, there exist $0<\beta_1<\beta_2<\infty$ such that for every $\beta\in(\beta_1,\beta_2)$ the loop sizes satisfy $P_{\beta,u}(|L(0)|=k)\le ae^{-bk}$ while the coupled bond percolation with $p=1-e^{-\beta}$ has an infinite cluster with positive probability (Corollary 2.2).
  • For sequences of $d$-regular expander graphs with diverging girth, macroscopic loops appear strictly later than macroscopic percolation clusters (Corollary 4.1).
  • The strict inequality holds for all $u\in(0,1]$, while the case $u=0$ remains open and appears to require new tools for dependent percolation (Section 4.4).
  • For graphs of diverging vertex degree, including complete graphs, hypercubes, and Hamming graphs, known results give equality $\beta_c=\beta_c^{\mathrm{per}}$, so the strict gap is specific to bounded degree (Conjecture 4.2 and the discussion around it).

Reading between the lines

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

  • The uniform domination produced by Proposition 3.1 is quantitative, so the same construction could in principle yield explicit lower bounds on the size of the gap $\beta_c(u)-\beta_c^{\mathrm{per}}$ on lattices, although the paper only proves positivity.
  • The red-edge event is an 'essential enhancement' with a built-in independence mechanism; this suggests that similar local cancellation events could be designed for other loop-weighted models once a comparison model is identified, a direction the paper mentions but does not develop beyond $\theta>1$ in an 'appropriate sense'.
  • The failure of the argument at $u=0$ points to a qualitative difference between two-cross cancellation (where two crosses on the same edge undo a connection) and double-bar dynamics, where no such cancellation exists; testing numerically whether the gap closes as $u\to0$ on $\mathbb{Z}^2$ would isolate which feature of the model is responsible for the strict inequality.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies random loop models on connected, countably infinite graphs of uniformly bounded degree, parametrised by a time parameter beta and a cross/double-bar intensity u. The main result, Theorem 2.1, asserts that for every u in (0,1] the critical loop parameter beta_c(u) is strictly larger than the critical percolation parameter beta_c^per defined by p=1-e^{-beta}; equivalently, there is an interval of beta where infinite percolation clusters occur almost surely while infinite loops do not. The proof constructs a coupled three-colour percolation process: red edges carry exactly two crosses and no neighbouring links between them, blue edges carry at least one link but are not red, and loops are contained in blue clusters. The central technical step, Proposition 3.1, claims that the red-edge process conditioned on all edges being coloured dominates a non-degenerate product Bernoulli measure with parameter delta depending only on beta and the degree bound. From this domination the authors derive that the blue process is subcritical for beta slightly above beta_c^per, yielding the theorem. The paper also contains a corollary for Z^d, an extension to expander graphs, and a discussion of why the u=0 case is not covered.

Significance. If the main theorem is correct, it establishes a strict separation between loop and Bernoulli percolation on all bounded-degree graphs, extending a result previously known for regular trees and providing the first general confirmation that the inequality beta_c >= beta_c^per can be strict outside mean-field settings. The proof idea is attractive and genuinely different from earlier tree arguments: it uses a stochastic domination criterion of Liggett-Schonmann-Stacey to show that the 'cancelling' red edges are sufficiently abundant and sufficiently independent. The claimed uniform dependence of delta only on beta and Delta is exactly the right kind of statement, and Corollary 2.2 is a concrete, falsifiable consequence for Z^d. The paper is self-contained in its derivation and does not disguise fitted parameters as predictions. However, as detailed below, a key inequality in the proof of the uniform domination is false, so the central argument is not yet valid as written.

major comments (3)
  1. [§3.4, Lemma 3.11, Eq. (3.17)] Inequality (3.17) is false. The text asserts that conditioning on type-1 neighbours of ~e only makes b_~e - a_~e smaller in probability, because X_~e becomes a Poisson point process on a subset of [0, beta] with the same intensity. But deleting an interval from the support can split the support into two components and thereby reduce the probability that two independent uniform points are within a fixed distance c. For beta=10 and c=1.25, two independent uniform points on [0,10] have range less than c with probability 1 - (1 - 1.25/10)^2 = 0.2344; on S = [0,4.9] ∪ [5.1,10] the same probability is 0.5(1 - (1 - 1.25/4.9)^2) ≈ 0.2225, which is strictly smaller. Thus conditioning on a type-1 neighbour can decrease, not increase, the left-hand side of (3.17). This inequality is the only justification for the uniform lower bound in Lemma 3.11, and Lemma 3.11 is used in the chain (3.18) to produce the positive constant in (3.19). The proof of Proposition 3.1, and therefore of Theorem 2.1, is not valid as written.
  2. [§3.2, reduction to u=1] The claim that it suffices to prove Proposition 3.1 for u=1 because u in (0,1) 'merely decreases the intensity of red edges by a factor u^2 > 0' is not established. Under the measure mu conditioned on n_e > 0, the probability of the red event depends on u through the Poisson intensity u, through the conditioning on at least one link, and through the fact that double bars on neighbouring edges also count in the condition N_~e(a_e,b_e] = 0. A simple factor u^2 does not obviously absorb all of these effects. Since Theorem 2.1 is stated for all u in (0,1], this reduction is load-bearing and requires a proof rather than a heuristic sentence.
  3. [§3.2, definition of mu; Proposition 3.1] The conditional measure mu = P_{beta,u}( . | n_e > 0 for all e in E') is not defined in the usual sense: for an infinite edge set E', the event {n_e > 0 for all e in E'} has probability zero under the product law, so conditioning on it requires a limiting or regular-conditional construction. Remark 3.2 acknowledges the issue and refers to discretisation of time and taking limits, but no such construction is actually given. Since Proposition 3.1 is used for infinite subgraphs in the proof of Theorem 2.1, this gap should be closed explicitly, for example by proving the domination first for finite subgraphs and then passing to the limit.
minor comments (4)
  1. [§3.2] The sentence 'cycles must be subsets of percolation clusters' uses 'cycles' where 'loops' is meant; the subsequent notation L(v) ⊆ C^B(v) is clear, but the terminology should be corrected.
  2. [Remark 3.2] The promise that conditioning on probability-zero events can be avoided by discretising time and taking limits is not followed by an actual limiting argument in the proof; this is related to major comment 3 and should be addressed.
  3. [Lemma 3.9] The proof by contradiction would be easier to verify if the dependence of epsilon on the outer configuration were made explicit; currently it is not fully clear why the two smallness conditions on the two terms in (3.9) cannot hold simultaneously for all epsilon > 0.
  4. [Eq. (3.13)] The notation 'η 3.9 ≤' in the displayed inequality is unclear; the intended use of Lemma 3.9 should be written as an explicit factor eta on the left-hand side.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from an explicit stochastic-domination argument, not from the definitions of the critical parameters.

full rationale

The paper's central claim, beta_c(u) > beta_c^per, is not equivalent to its inputs by construction. The lower bound beta_c >= beta_c^per is obtained from the natural coupling between loop configurations and Bernoulli bond percolation, where an edge is open exactly when it carries at least one link; this is a genuine comparison between two separately defined critical parameters. The strict inequality is then proved by constructing red edges whose removal stochastically dominates deletion by an independent Bernoulli thinning: Proposition 3.1 establishes that, conditioned on all edges carrying at least one link, the red-edge process dominates a product Bernoulli measure with a uniform delta > 0. The proof of Proposition 3.1 proceeds through local lemmas about Poisson point processes on individual edges and their neighbours, using a spatial Markov property and finite boundary conditions; no parameter is fitted to data, and no quantity called a prediction is in fact an input. The paper's citations, including [LSS97] for product-measure domination and [Ham15] for the tree case, are used as external tools or benchmarks rather than as the sole justification of the main theorem. A referee might dispute the validity of inequality (3.17) in Lemma 3.11, but that would be a mathematical correctness gap, not circularity: the lemma is asserted and proved, not assumed as the conclusion. Since no self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation is present, the appropriate circularity score is 0.

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

The central proof introduces no fitted numeric parameters; the constants delta, eta, N, and epsilon' are existence constants for given beta and Delta, not fitted to data. It relies on standard stochastic domination and Poisson process facts, the model definition, and the assumption that cancellation of two crosses on an edge makes that edge unavailable to loops. The red-edge construction is a proof device rather than a postulated physical entity, so invented_entities is empty.

assumptions (5)
  • standard math Liggett-Schonmann-Stacey domination criterion: a process dominates a product Bernoulli measure if conditional probabilities of success are uniformly bounded below.
    Invoked around equation (3.3) to reduce Proposition 3.1 to verifying (3.3); used as an external theorem [LSS97].
  • domain assumption Loop clusters are contained in blue clusters, i.e. L(v) subset C^B(v) in equation (3.1).
    This cancellation assertion is the bridge between loop finiteness and blue percolation; the paper justifies it with transposition involution and commutativity but does not give a full proof.
  • standard math beta_c is well-defined and independent of the reference vertex, following Ang03 Proposition 5.
    Used to define beta_c in equation (2.3) and to state Theorem 2.1.
  • standard math For p > p_c, Bernoulli bond percolation has an infinite cluster almost surely.
    Standard tail-event argument; used in Corollary 2.2 and the abstract's 'almost surely' phrasing.
  • domain assumption Graphs are connected, countably infinite, uniformly bounded degree, with p_c < 1.
    Explicit hypotheses of Theorem 2.1; p_c < 1 excludes one-dimensional chains where the statement degenerates to infinity = infinity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Critical Parameters for Loop and Bernoulli Percolation." pith.science (2026). https://pith.science/paper/4BECDABH

@misc{pith2026190810213,
  author       = {Pith},
  title        = {Pith review of: Critical Parameters for Loop and Bernoulli Percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4BECDABH}},
  note         = {Machine review of arXiv:1908.10213}
}
abstract

We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $\beta\geq 0$. Intuitively, larger $\beta$ means more randomness. In particular, at $\beta=0$ we start with loops of length 1 and as $\beta$ crosses a critical value $\beta_c$, infinite loops start to occur almost surely. Our random loop models admit a natural comparison to bond percolation with $p=1-e^{-\beta}$ on the same graph to obtain a lower bound on $\beta_c$. For those graphs of diverging vertex degree where $\beta_c$ and the critical parameter for percolation have been calculated explicitly, that inequality has been found to be an equality. In contrast, we show in this paper that for graphs of bounded degree the inequality is strict, i.e. we show existence of an interval of values of $\beta$ where there are no infinite loops, but infinite percolation clusters almost surely.

Figures

Figures reproduced from arXiv: 1908.10213 by the authors.

Figure 1
Figure 1. Graphs and realizations of Poisson point processes, and their loops. In both cases, there are exactly two loops, one in red and one in blue. In Section 2 we introduce some more notation and state our main result, Theorem 2.1, rigorously. In Section 3 we prove our main result. In Section 4 we discuss briefly some natural [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. An example of a configuration on three edges e1, e2, e3. All edges are coloured (Se1 = Se2 = Se3 = 1) because ne1 = ne2 = ne3 = 2 > 0. e1 is red (Re1 = 1) since e1 has two crosses and no neighbours with a link between ae1 and be1 . e2 is blue (Be2 = 1) because ne2 > 0, but not both links are crosses. e3 is blue too because it has a neighbour, e2, with a link between its two crosses, i.e. Ne2 (ae3 , be3 ] > 0. Note t… view at source ↗
Figure 3
Figure 3. Examples for e0 being pivotal for f = Re˜, i.e. Pe˜ (left) and P c e˜ (right). The shaded region indicates that Xe0 is not conditioned on, i.e. is still random. Note that conditioning on Re˜ being equal to 0 (or 1) will make ˜e a type 0 (or a type 1) neighbour of e0 in the left picture. In the right picture conditioning on Re˜ = 1 is not admissible. Proof. Note that it suffices to prove that µ [PITH_FULL_IMAGE:figu… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The left picture illustrates how the probability of e0 being red is small if there are many crosses on its neighbours, while the right picture illus￾trates what could go wrong with type 1 neighbours. The area shaded in red emphasises that, in order not to violate Re˜ =…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 41 canonical work pages

  1. [1]

    Percolation on finite graphs and isoperimetric inequalities

    Noga Alon, Itai Benjamini, Alan Stacey, et al. Percolation on finite graphs and isoperimetric inequalities. The Annals of Probability , 32(3):1727--1745, 2004

    work page 2004

  2. [2]

    Strict monotonicity for critical points in percolation and ferromagnetic models

    Michael Aizenman and Geoffrey Grimmett. Strict monotonicity for critical points in percolation and ferromagnetic models. Journal of Statistical Physics , 63(5-6):817--835, 1991

    work page 1991

  3. [5]

    Geometric aspects of quantum spin states

    Michael Aizenman and Bruno Nachtergaele. Geometric aspects of quantum spin states. Communications in Mathematical Physics , 164(1):17--63, 1994

    work page 1994

  4. [6]

    Random infinite permutations and the cyclic time random walk

    Omer Angel. Random infinite permutations and the cyclic time random walk. Discrete Mathematics and Theoretical Computer Science AC , pages 9--16, 2003

    work page 2003

  5. [7]

    A numerical study of the 3d random interchange and random loop models

    Alessandro Barp, Edoardo Gabriele Barp, Fran c ois-Xavier Briol, and Daniel Ueltschi. A numerical study of the 3d random interchange and random loop models. Journal of Physics A: Mathematical and Theoretical , 48(34):345002, 2015

    work page 2015

  6. [8]

    Sharp threshold for percolation on expanders

    Itai Benjamini, St \'e phane Boucheron, G \'a bor Lugosi, Rapha \"e l Rossignol, et al. Sharp threshold for percolation on expanders. The Annals of Probability , 40(1):130--145, 2012

    work page 2012

  7. [9]

    Essential enhancements revisited

    Paul Balister, B \'e la Bollob \'a s, and Oliver Riordan. Essential enhancements revisited. arXiv preprint arXiv:1402.0834 , 2014

    arXiv 2014

  8. [11]

    Emergence of giant cycles and slowdown transition in random transpositions and k -cycles

    Nathana \"e l Berestycki. Emergence of giant cycles and slowdown transition in random transpositions and k -cycles. Electronic Journal of Probability , 16:152--173, 2010

    work page 2010

Show all 43 references
  1. [13]

    Large cycles in random permutations related to the heisenberg model

    Jakob Bj \"o rnberg. Large cycles in random permutations related to the heisenberg model. Electronic Communications in Probability , 20, 2015

    2015

  2. [14]

    The free energy in a class of quantum spin systems and interchange processes

    Jakob Bj \"o rnberg. The free energy in a class of quantum spin systems and interchange processes. Journal of Mathematical Physics , 57(7):073303, 2016

    2016

  3. [16]

    Critical parameter of random loop model on trees

    Jakob Bj \"o rnberg and Daniel Ueltschi. Critical parameter of random loop model on trees. The Annals of Applied Probability , 28(4):2063--2082, 2018

    2018

  4. [18]

    Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction

    Sacha Friedli and Yvan Velenik. Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction . Cambridge University Press, 2017

    2017

  5. [19]

    Infinite cycles in the random stirring model on trees

    Alan Hammond. Infinite cycles in the random stirring model on trees. Bulletin of the Institute of Mathematics Academia Sinica , 8:85--104, 2012

    2012

  6. [20]

    Sharp phase transition in the random stirring model on trees

    Alan Hammond. Sharp phase transition in the random stirring model on trees. Probability Theory and Related Fields , 161(3-4):429--448, 2015

    2015

  7. [21]

    Theodore E. Harris. Nearest-neighbor markov interaction processes on multidimensional lattices. Advances in Mathematics , 9(1):66--89, 1972

    1972

  8. [23]

    The random interchange process on the hypercube

    Roman Koteck \'y , Piotr Mi o \'s , and Daniel Ueltschi. The random interchange process on the hypercube. Electronic Communications in Probability , 21:1--9, 2016

    2016

  9. [24]

    Domination by product measures

    Thomas M Liggett, Roberto H Schonmann, and Alan M Stacey. Domination by product measures. The Annals of Probability , 25(1):71--95, 1997

    1997

  10. [26]

    Compositions of random transpositions

    Oded Schramm. Compositions of random transpositions. Israel Journal of Mathematics , 147(1):221--243, 2005

    2005

  11. [27]

    Improved lower bound on the thermodynamic pressure of the spin 1/2 heisenberg ferromagnet

    B \'a lint T \'o th. Improved lower bound on the thermodynamic pressure of the spin 1/2 heisenberg ferromagnet. Letters in Mathematical Physics , 28(1):75--84, 1993

    1993

  12. [28]

    Random loop representations for quantum spin systems

    Daniel Ueltschi. Random loop representations for quantum spin systems. Journal of Mathematical Physics , 54(8), 2013

    2013

  13. [29]

    Harris, T. E. Nearest-neighbor Markov interaction processes on multidimensional lattices. Advances in Mathematics 9.1 (1972): 66-89

    1972

  14. [30]

    Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet

    T\' o th, B. Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet. letters in mathematical physics 28.1 (1993): 75-84

    1993

  15. [31]

    and Nachtergaele, B

    Aizenman, M. and Nachtergaele, B. Geometric aspects of quantum spin states. Communications in Mathematical Physics 164.1 (1994): 17-63

    1994

  16. [32]

    Random loop representations for quantum spin systems

    Ueltschi, D. Random loop representations for quantum spin systems. Journal of Mathematical Physics 54.8 (2013): 083301

    2013

  17. [33]

    Domination by product measures

    Liggett, Thomas M., Schonmann, Roberto H., and Stacey, Alan M. Domination by product measures. The Annals of Probability 25.1 (1997): 71-95

    1997

  18. [34]

    Statistical mechanics of lattice systems: a concrete mathematical introduction

    Friedli, Sacha, and Yvan Velenik. Statistical mechanics of lattice systems: a concrete mathematical introduction. Cambridge University Press, 2017

    2017

  19. [35]

    and Barp, E.G

    Barp, A. and Barp, E.G. and Briol, F.X. and Ueltschi, D. A numerical study of the 3D random interchange and random loop models. Journal of Physics A: Mathematical and Theoretical 48.34 (2015): 345002

    2015

  20. [36]

    and Kotowski, M

    Bj\" o rnberg, J.E. and Kotowski, M. and Lees, B. and Mi o\' s , P. The interchange process with reversals on the complete graph. Preprint. arXiv:1812.03301 (2018)

    2018 arXiv

  21. [37]

    Compositions of random transpositions

    Schramm, O. Compositions of random transpositions. Israel Journal of Mathematics 147.1 (2005): 221-243

    2005

  22. [38]

    and Mi o\' s , P

    Koteck\' y , R. and Mi o\' s , P. and Ueltschi, D. The random interchange process on the hypercube. Electron. Commun. Probab. 21 (2016): 1-9

    2016

  23. [39]

    Random Infinite Permutations and the Cyclic Time Random Walk

    Angel, O. Random Infinite Permutations and the Cyclic Time Random Walk. DRW. 2003

    2003

  24. [40]

    Infinite cycles in the random stirring model on trees

    Hammond, A. Infinite cycles in the random stirring model on trees. Bulletin of the Institute of Mathematics Academia Sinica 8 (2012): 85-104

    2012

  25. [41]

    Sharp phase transition in the random stirring model on trees

    Hammond, A. Sharp phase transition in the random stirring model on trees. Probability Theory and Related Fields 161.3-4 (2015): 429-448

    2015

  26. [42]

    and Ueltschi, D

    Bj\" o rnberg, J.E. and Ueltschi, D. Critical parameter of random loop model on trees. The Annals of Applied Probability 28.4 (2018): 2063-2082

    2018

  27. [43]

    and Hegde, M

    Hammond, A. and Hegde, M. Critical point for infinite cycles in a random loop model on trees. Preprint. arXiv:1805.11772 (2018)

    2018 arXiv

  28. [44]

    and Mi o\' s , P

    S eng\" u l, B. and Mi o\' s , P. Existence of a phase transition of the interchange process on the Hamming graph. Preprint. arXiv:1605.03548 (2016), accepted to EJP

    2016 arXiv

  29. [45]

    Large cycles in random permutations related to the Heisenberg model

    Bj\" o rnberg, J.E. Large cycles in random permutations related to the Heisenberg model. Electronic Communications in Probability 20 (2015)

    2015

  30. [46]

    The free energy in a class of quantum spin systems and interchange processes

    Bj\" o rnberg, J.E. The free energy in a class of quantum spin systems and interchange processes. Journal of Mathematical Physics 57.7 (2016): 073303

    2016

  31. [47]

    and Kozma, G

    Alon, G. and Kozma, G. The mean-field quantum Heisenberg ferromagnet via representation theory. Preprint. arXiv:1811.10530 (2018)

    2018 arXiv

  32. [48]

    o rnberg, J.E. and Fr\

    Bj\" o rnberg, J.E. and Fr\" o hlich, J. and Ueltschi, D. Quantum spins and random loops on the complete graph. Preprint. arXiv:1811.12834 (2018)

    2018 arXiv

  33. [49]

    and Ueltschi, D

    Bj\" o rnberg, J.E. and Ueltschi, D. Critical temperature of Heisenberg models on regular trees, via random loops. Preprint. arXiv:1803.11430 (2018)

    2018 arXiv

  34. [50]

    and Ehlert, J

    Betz, V. and Ehlert, J. and Lees, B. Phase transition for loop representations of Quantum spin systems on trees. Preprint. arXiv:1804.00860 (2018)

    2018 arXiv

  35. [51]

    and Kotowski, M

    Adamczak, R. and Kotowski, M. and Mi o\' s , P. Phase transition for the interchange and quantum Heisenberg models on the Hamming graph. Preprint arXiv:1808.08902 (2018)

    2018 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.