Some inequalities for reversible Markov chains and branching random walks via spectral optimization

Jonathan Hermon

REVIEW 1 major objections 4 minor 25 references

Some inequalities for reversible Markov chains and branching random walks via spectral optimization

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves a quantitative inequality showing that a reversible chain's uniform mixing time is at most its relaxation time times the logarithm of the ratio of hitting to relaxation time, and uses it to resolve the Aldous–Fill…

desk verdict Sharp spectral inequality relating mixing and hitting times, with a proof that holds up; the Aldous–Fill resolution is real but rests on Oliveira as advertised. read the letter →

arxiv 1908.08525 v4 pith:KLGDV6RO submitted 2019-08-22 math.PR

Jonathan Hermon This is my paper · ORCID

classification math.PR MSC 60J1060J2760J8060K3505C81

keywords mixingtimeshittingspectraloptimizationbranchingrandomwalkintersectiongapvertex-transitivegraphscoalescing

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

For any finite reversible Markov chain, the paper bounds the uniform ($L_\infty$) mixing time in terms of two more basic quantities: the relaxation time $t_{\mathrm{rel}} = 1/\mathrm{gap}$ and the maximal expected hitting time of a state $t_{\mathrm{hit}}$. The main inequality, $t_{\mathrm{mix}}^{(\infty)} \le C t_{\mathrm{rel}}\log(1 + t_{\mathrm{hit}}/t_{\mathrm{rel}})$, combined with the classical reverse direction, characterizes exactly when uniform mixing and hitting times separate. This completes a proof, begun in an earlier theorem cited in the paper, that under vertex-transitivity the spectral condition $t_{\mathrm{rel}} \ll t_{\mathrm{hit}}$ forces mean-field behavior for coalescing random walks. The same method yields higher-order versions and a branching-random-walk reading: the maximal expected hitting time, and under transitivity the expected intersection time, of a branching random walk whose particles split at rate $\mathrm{gap}$, are comparable to the same logarithmic quantities.

What carries the argument

The engine is a 'spectral optimization' principle. The $L_\infty$ mixing time is controlled by a sum of exponentials $\sum_i e^{-\lambda_i t}$ involving the eigenvalues $\lambda_i$ of $I - P$, while the expected hitting time $t_{\pi \to x}$ and the average hitting time $t_\odot$ are the same spectral sums with $1/\lambda_i$ weights. The paper maximizes the exponential sum subject to the constraint that the weighted sum of $1/\beta_i^\ell$ is held fixed, and proves that the maximum is attained by concentrating all spectral weight at the smallest eigenvalue $\lambda_2 = \mathrm{gap}$, with the remaining weights sent to $\infty$. The key monotonicity fact is that $x^\ell e^{-2xt}$ decreases for $x \ge 1/(2t)$ when $t \ge \ell t_{\mathrm{rel}}/2$, so replacing any larger $\beta_i$ by mass at $\lambda_2$ preserves the constraint and only increases the objective. This reduces a mixing-time estimate to evaluating one exponential, and generalizes to $\ell$-th order quantities $Q_\ell = \sum_{i \ge 2} \lambda_i^{-\ell}$ and $\sigma_{x,\ell}$, yielding the higher-order and intersection-time bounds.

What would settle it

A concrete refutation would be a finite irreducible reversible chain whose exact diagonalization gives $t_{\mathrm{mix}}^{(\infty)}(1/2) > t_{\mathrm{rel}}\max\{1,\log(2\max_x t_{\pi \to x}/t_{\mathrm{rel}})\}$; searching small birth-and-death chains by exact computation is the direct test of Theorem 1.

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Extended reading notes

Core claim

The central discovery is Theorem 1: for every irreducible reversible Markov chain on a finite state space and every $\varepsilon > 0$, $t_{\mathrm{mix}}^{(\infty)}(\varepsilon) \le t_{\mathrm{rel}}\max\{1,\log(\max_x t_{\pi \to x}/(\varepsilon t_{\mathrm{rel}}))\}$, and in particular $t_{\mathrm{mix}}^{(\infty)} \lesssim t_{\mathrm{rel}}\log(1 + t_{\mathrm{hit}}/t_{\mathrm{rel}})$. Because the classical bound $t_{\mathrm{mix}}^{(\infty)} \ge t_{\mathrm{rel}}\log 2$ goes in the opposite direction, the paper obtains a two-sided spectral characterization: a sequence of finite reversible chains has uniform mixing time of strictly smaller order than the maximal hitting time if and only if the relaxation time is of strictly smaller order than the hitting time, with the analogous equivalence for constancy up to factors. The paper then combines this with an earlier theorem, quoted as [20], that on vertex-transitive graphs total-variation mixing time much smaller than hitting time already implies mean-field coalescence, thereby resolving the Aldous–Fill conjecture (Open Problem 14.12). In the same framework, the expected hitting time of a critical branching random walk—where each particle splits at rate $\mathrm{gap}$—is shown to be comparable to $t_{\mathrm{rel}}\log(1 + t_{\mathrm{hit}}/t_{\mathrm{rel}})$, and in the transitive case the expected intersection time of two such walks is comparable to $t_{\mathrm{rel}}\log(1 + \sqrt{Q}/t_{\mathrm{rel}})$.

Load-bearing premise

The load-bearing premise is that the earlier theorem quoted as [20] is correct: on vertex-transitive graphs, total-variation mixing much smaller than hitting already forces mean-field coalescence, with the new paper supplying the equivalence that turns the spectral condition into that mixing-time condition.

Editorial extensions

If this is right

For a sequence of reversible finite-state chains, $t_{\mathrm{mix}}^{(\infty)} \ll t_{\mathrm{hit}}$ holds if and only if $t_{\mathrm{rel}} \ll t_{\mathrm{hit}}$, so the spectral-gap–hitting-time product is a complete criterion for separation of the two time scales.
The Aldous–Fill conjecture holds: on vertex-transitive graphs, $t_{\mathrm{rel}} \ll t_{\mathrm{hit}}$ implies that the coalescence time of coalescing random walks, rescaled by the meeting time, converges to the Kingman-coalescent law.
A branching random walk in which each particle splits at rate $\mathrm{gap}$ has maximal expected hitting time comparable to $t_{\mathrm{rel}}\log(1 + t_{\mathrm{hit}}/t_{\mathrm{rel}})$, and under transitivity its expected intersection time is comparable to $t_{\mathrm{rel}}\log(1 + \sqrt{Q}/t_{\mathrm{rel}})$, refining earlier intersection-mixing bounds.
The condition $t_{\mathrm{mix}}^{\mathrm{TV}} \ll t_{\mathrm{hit}}$, and its $L_\infty$ analogue, is stable under rough isometries and small edge-weight perturbations, because both are equivalent to the robust spectral condition $t_{\mathrm{rel}} \ll t_{\mathrm{hit}}$.
Higher-order versions recover sharp mixing-time estimates on tori: for the $d$-dimensional torus, $t_{\mathrm{mix}}^{(\infty)}(\mathbb{Z}_m^d) = O(d\,t_{\mathrm{rel}})$, matching the true order up to a dimension-dependent constant.

Reading between the lines

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

The spectral-optimization mechanism is not tied to $\ell = 1$; it suggests that for $\ell \ge 3$ the quantities $Q_\ell = \sum_{i \ge 2} \lambda_i^{-\ell}$ may admit probabilistic interpretations as multi-particle intersection times, which the paper leaves open.
If the method extends to infinite reversible chains with spectral radius $\rho$ replacing the spectral gap, the branching-random-walk picture would give a natural critical-branching criterion for mixing behavior on infinite graphs.
A practical consequence of the branching-random-walk bounds is a Monte-Carlo route to mixing-time estimates: simulate two branching random walks with splitting rate $\mathrm{gap}$, estimate their intersection time, and read off $t_{\mathrm{rel}}\log(1 + \sqrt{Q}/t_{\mathrm{rel}})$, without computing eigenvalues.
The equivalence $t_{\mathrm{mix}}^{(\infty)} \ll t_{\mathrm{hit}} \iff t_{\mathrm{rel}} \ll t_{\mathrm{hit}}$ suggests that families of graphs with few small Laplacian eigenvalues generically have mixing time of smaller order than hitting time, because the spectral-optimization worst case requires mass concentration near the edge of the spectrum.

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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

Short version: this is a real result, not a repackaging. The new inequality t_mix^∞(ε) ≤ trel max{1, log(max_x tπ→x/(ε trel))} and Corollary 1.1 are proved in full; the spectral optimization argument is fresh and the proof of Theorem 2 holds up on inspection. The paper's value is in giving a sharp, parameter-free relation between trel, thit, and t_mix^∞, and it genuinely closes the equivalence that the Aldous–Fill conjecture needed.

What is new: the ℓ=1 bound and its corollary are new, as is the higher-order ℓ≥2 family (Theorem 2), the sharp torus example, and the branching random walk interpretation. The proof is honest: no fitted constants, no circular stepping. The case analysis in Theorem 2 rules out non-extremal configurations, and the spectral identities used are standard. I did not find a load-bearing gap.

Soft spots: the headline resolution of Aldous–Fill is explicitly built on Oliveira [20]. If Oliveira's theorem had a gap or different hypotheses, the conjecture resolution would fail even though Theorem 1 stands. That is a normal division of labor, but the referee should check the exact statement in [20] and verify the chain conventions. The paper should also label the equivalence more carefully: in the introduction, the sentence saying 'Theorem 3 asserts that this condition is equivalent' is wrong as printed, since Theorem 3 concerns BRW hitting times; it should point to Corollary 1.1 or Theorem 1. Minor presentational issues: the derivation of (4.11) is omitted as routine, and in Lemma 4.3 H_s(x,x) is called non-decreasing when it should be decreasing. Neither affects the main conclusions.

Bottom line: the central argument holds, and the paper deserves a serious referee. I would accept after minor revisions. It is worth reading and citing if you work on mixing times, hitting times, or coalescing random walks.

Referee Report

1 major / 4 minor

Summary. The paper develops a spectral-optimization method for reversible finite-state Markov chains and uses it to prove new quantitative relations between the L∞ mixing time, maximal hitting time, relaxation time, and higher-order spectral quantities. Theorem 1 gives t_mix^(∞)(ε) ≤ t_rel max{1, log(max_x t_{π→x}/(ε t_rel))}, hence t_mix^(∞) ≲ t_rel log(1 + t_hit/t_rel), and Corollary 1.1 characterizes when uniform mixing times and maximal hitting times are of the same order in terms of t_rel versus t_hit. Theorem 2 extends the method to higher ℓ and to average/pointwise L2 mixing times. Theorems 3 and 4 interpret these bounds through branching random walks whose particle number grows at rate of the spectral gap, with transitive chains giving a connection between intersection times of two BRWs and the L∞ mixing time. The paper further claims that this resolves Aldous and Fill's Open Problem 14.12 on mean-field coalescence, with the heavy lifting attributed to Oliveira [20].

Significance. If the results stand, Theorem 1 is a strong parameter-free improvement over the classical bounds (1.4), and Corollary 1.1 is a clean spectral characterization of the comparison t_mix^(∞) ≍ t_hit versus t_rel ≍ t_hit. The spectral-optimization proof is novel and appears to be complete; the BRW interpretation is a genuine probabilistic consequence rather than a restatement of the inequalities. The constants are universal and no fitted parameters appear. The main caveat is that the headline 'resolves the Aldous–Fill conjecture' is conditional on Oliveira [20], which the paper does not re-derive. In addition, Theorem 4 contains a displayed inequality with a factor error that must be corrected before publication; the qualitative '≲' form of the transitive statement remains correct after the fix.

major comments (1)

[§1.2, Eq. (1.16)] The displayed inequality t_mix^(∞) ≤ t_rel max{1, (1/2) log(4Q/t_rel^2)} is false as stated. For the rate-1 reversible walk on the complete graph K_n, t_rel ≈ 1, Q ≈ n, and with the paper's convention t_mix^(∞) = t_mix^(∞)(1/2) we have t_mix^(∞) ≈ log(2n), whereas the right-hand side is about (1/2)log(4n); for large n the inequality fails by a factor of roughly 2 in the leading logarithm. The sentence 'the inequality is immediate from (1.6)' is consistent with the correct bound t_mix^(∞)(1/2) ≤ t_rel max{2, log(2Q/t_rel^2)}, or equivalently the bound t_mix^(∞)(1/4) ≤ t_rel max{2, log(4Q/t_rel^2)} already stated in §1.1. Please correct (1.16) and its proof; the later qualitative estimate t_mix^(∞) ≲ t_rel log(1 + √Q/t_rel) is not affected.

minor comments (4)

[§1, p. 6] The sentence 'However, Theorem 3 asserts that this condition is in fact equivalent to the condition t_rel^(n) ≪ t_hit^(n)' is a cross-reference error: Theorem 3 concerns branching random walk hitting times, not the equivalence t_TV_mix ≪ t_hit ⇔ t_rel ≪ t_hit. That equivalence is Corollary 1.1 together with (2.4). Similarly, §1.1 refers to 'Theorem 3 (namely (1.16))', but (1.16) appears in Theorem 4.
[Lemma 4.3 proof] The proof states that H_s(x,x) is non-decreasing in s; earlier in the paper and by spectral theory it is decreasing. The subsequent integral comparison is still valid, but the monotonicity statement should be corrected to avoid confusing readers.
[Lemma 4.3 proof] The chain 't_rel ≤ √Q ≲ ∑_x π(x)ρ_x ≤ ρ_max' is dimensionally inconsistent and is not what the proof needs. The needed inequality is t_rel^2 ≤ Q ≲ ∑_x π(x)ρ_x ≤ ρ_max, which follows from (1.9)–(1.10). Please fix this display.
[Transitive part of Theorem 4] The proof of (4.11) asserts that P_{π,π}[τ_I > j(2E_{π,π}[τ_I] + t_mix^(∞)(1/4))] decays exponentially in j and says the details are routine. Since this exponential decay is used to obtain E_{π,π}[τ_I^2] ≲ (E_{π,π}[τ_I])^2, a short justification should be included.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spectral-optimization inequalities are derived from independently defined spectral quantities, and the only load-bearing external input (Oliveira's theorem) is a separate dependency, not a self-reference or a constructed equivalence.

full rationale

The central derivations are self-contained and non-circular. Theorem 2 is proved by spectral decomposition: the quantities Q_l = sum_i lambda_i^{-l} and sigma_{x,l} = sum_i f_i(x)^2/lambda_i^l are independently defined from the eigenvalues/eigenfunctions, and the identities sigma_{x,1} = t_{pi->x} and Q_1 = t_odot are exact classical spectral identities, not assumptions of the conclusion. The optimization problem maximizes sum a_i e^{-2 beta_i t} subject to sum a_i beta_i^{-l} = Q_l, and the proof shows the maximum is attained at beta_1 = lambda_2, a_1 = Q_l lambda_2^l by an elementary monotonicity argument for h(x) = x^l e^{-2xt}; no fitted parameter is renamed as a prediction. Theorem 1 is then the l=1 case of Theorem 2, with the right-hand side trel log(1 + thit/trel) built from trel and thit, and Corollary 1.1 follows from (1.1) together with the classical lower bound trel log 2 <= t_mix^{(infinity)}; the equivalence is a consequence, not a definition. The branching random walk statements are interpretations/consequences rather than inputs: the paper explicitly says the first inequality in (1.12) is 'a repetition of (1.1)', and the matching BRW estimates are proved separately by Paley-Zygmund and Markov arguments, so the BRW hitting and intersection times do not feed back into Theorem 1. The Aldous-Fill resolution does rely on Oliveira [20] for the mean-field behavior of coalescing random walks, but this is an acknowledged external theorem, structurally separate from the new inequality, and it is not a self-citation; the paper states 'Oliveira [20] has already done all of the heavy lifting'. Self-citations in the paper ([6], [11], [13], [15], [16]) are contextual, peripheral, or used for secondary robustness remarks, and none is load-bearing for Theorem 1 or Corollary 1.1. The possible typo in Lemma 4.3 ('non-decreasing' where monotonicity in the intended direction is used) is a presentation slip and does not create a circular step. Overall, no prediction or claimed first-principles result reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted and no new physical entities are introduced. The branching random walk is a standard stochastic process used as a tool, with split rate set to the existing spectral gap. The axioms are standard spectral theory for finite reversible chains plus one external theorem from Oliveira [20].

assumptions (4)

standard math Spectral decomposition identity: H_t(x,x) minus pi(x) equals sum_i f_i(x)^2 e^{-lambda_i t}, and t_{pi to x} equals sum_i c_i(x)/lambda_i with the same coefficients c_i(x).
Invoked in Section 1.3 and Section 2.2, Eq. (2.6); this identity makes the optimization constraint match the hitting-time sum and is load-bearing for the main inequality.
standard math Heat-kernel exponential tail bound (2.10): H_{t+s}(x,x) minus pi(x) is at most e^{-s/trel} times H_t(x,x) minus pi(x).
Used in Lemma 2.2 to truncate infinite integrals to finite windows; requires reversibility and controls the cost of truncation.
standard math Random target identity and eigentime identity: t_odot is independent of the starting state and equals sum_{i>=2} 1/lambda_i.
Used in Section 2.2 and in the optimization cutoff choice; connects average hitting time to the spectral sum.
domain assumption Oliveira's mean-field coalescence theorem for vertex-transitive graphs under t_TV_mix much smaller than thit.
External theorem cited as the heavy lifting; Corollary 1.1 makes its hypothesis equivalent to trel much smaller than thit, so the conjecture resolution is conditional on this published result.

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Pith. "Pith review of Some inequalities for reversible Markov chains and branching random walks via spectral optimization." pith.science (2026). https://pith.science/paper/KLGDV6RO

@misc{pith2026190808525,
  author       = {Pith},
  title        = {Pith review of: Some inequalities for reversible Markov chains and branching random walks via spectral optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLGDV6RO}},
  note         = {Machine review of arXiv:1908.08525}
}

abstract

We present results relating mixing times to the intersection time of branching random walk (BRW) in which the logarithm of the expected number of particles grows at rate of the spectral-gap $\mathrm{gap}$ . This is a finite state space analog of a critical branching process. Namely, we show that the maximal expected hitting time of a state by such a BRW is up to a universal constant larger than the $L_{\infty}$ mixing-time, whereas under transitivity the same is true for the intersection time of two independent such BRWs. Using the same methodology, we show that for a sequence of reversible Markov chains, the $L_{\infty}$ mixing-times $t_{\mathrm{mix}}^{(\infty)} $ are of smaller order than the maximal hitting times $t_{\mathrm{hit}}$ iff the product of the spectral-gap and $t_{\mathrm{hit}}$ diverges, by establishing the inequality $t_{\mathrm{mix}}^{(\infty)} \le \frac{1}{\mathrm{gap}}\log(et_{\mathrm{hit}} \cdot \mathrm{gap}) $. This resolves a conjecture of Aldous and Fill (Reversible Markov chains and random walks on graphs, Open Problem 14.12) asserting that under transitivity the condition that $ t_{\mathrm{hit}} \gg \frac{1}{\mathrm{gap}} $ implies mean-field behavior for the coalescing time of coalescing random walks.

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Works this paper leans on

25 extracted references · 24 canonical work pages

[20]

Oliveira, R., Mean ﬁeld conditions for coalescing random walks. Ann. Probab. 41 (2013), no. 5, 3420–3461. MR3127887

work page 2013
[1]

Aldous, D., Hitting times for random walks on vertex-transitive gr aphs. Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 1, 179–191. MR0994089

work page 1989
[2]

Aldous, D., Mixing times and hitting times. (2010). Available at http://www.stat.berkeley. edu/ aldous/Talks/slides.html

work page 2010
[3]

Journal of the London Mathematical Society, 2(3):564–576, 1982

Aldous, D., Some inequalities for reversible Markov chains. Journal of the London Mathematical Society, 2(3):564–576, 1982. MR657512

work page 1982
[4]

Aldous, D., Threshold limits for cover times. J. Theoret. Probab. 4 (1991), no. 1, 197–211. MR1088401

work page 1991
[5]

Unﬁnished manuscript

Aldous, D., and Fill, J., Reversible Markov chains and random walks on graphs . Unﬁnished manuscript. Available at the ﬁrst author’s website

work page
[6]

Basu, R., Hermon, J., and Peres, Y., Characterization of cutoﬀ f or reversible Markov chains. Ann. Probab. 45 (2017), no. 3, 1448–1487. MR3650406

work page 2017
[7]

Y., and Saloﬀ-Coste, L., The cutoﬀ phenomenon for erg odic Markov processes

Chen, G. Y., and Saloﬀ-Coste, L., The cutoﬀ phenomenon for erg odic Markov processes. Electron. J. Probab. 13 (2008), no. 3, 26–78. MR2375599

work page 2008

Show all 25 references

[8]

Sensitivity of mixing times

Ding, J., and Peres, Y. Sensitivity of mixing times. Electron. Commun. Probab. 18 (2013), paper 88, 6pp. Available at: projecteuclid/1465315627

2013
[9]

Total variation cutoﬀ in birt h-and-death chains

Ding, J., Lubetzky, E., and Peres, Y. Total variation cutoﬀ in birt h-and-death chains. Probab. Theory Related Fields 146 (2010), no. 1-2, 61–85. MR2550359

2010
[10]

Markov Process

Gantert, N., and M¨ uller, S., The critical branching Markov chain is transient. Markov Process. Related Fields 12 (2006), no. 4, 805–814. MR2284404

2006
[11]

(2019) To appear in Journal of Theoretical Probab

Hermon, J., A spectral characterization for concentration o f the cover time. (2019) To appear in Journal of Theoretical Probab. Arxiv preprint arXiv:1809.00145

2019 arXiv
[12]

ALEA Lat

Hermon, J., A technical report on hitting times, mixing and cutoﬀ . ALEA Lat. Am. J. Probab. Math. Stat. 15 (2018), no. 1, 101–120. MR3765366

2018
[13]

On sensitivity of uniform mixing times

Hermon, J. On sensitivity of uniform mixing times. Ann. Inst. Henri Poincar´ e Probab. Stat. 54 (2018), no. 1, 234–248. Available at: projecteuclid/1519030827

2018
[14]

Arxiv preprint arXiv:2008.07517

Hermon, J., and Kozma, G., Sensitivity of mixing times of Cayley gra phs. Arxiv preprint arXiv:2008.07517

2008 arXiv
[15]

Hermon, J., and Peres, Y., A characterization of L2 mixing and hypercontractivity via hitting times and maximal inequalities. Probab. Theory Related Fields 170 (2018), no. 3-4, 769–800. MR3773799 26

2018
[16]

Electron

Hermon, J., and Peres, Y., On sensitivity of mixing times and cutoﬀ . Electron. J. Probab. 23 (2018), Paper No. 25, 34 pp. MR3779818

2018
[17]

Applie d Mathematical Sciences, 28

Keilson, J., Markov chain models -rarity and exponentiality. Applie d Mathematical Sciences, 28. Springer-Verlag, New York-Berlin, 1979. xiii+184 pp. ISBN: 0-387- 90405-0 MR0528293

1979
[18]

Markov chains and mixing times

Levin, D., and Peres, Y., (2017). Markov chains and mixing times. American Mathematical Society, Providence, RI. With contributions by Elizab eth L. Wilmer and a chapter by James G. Propp and David B. Wilson. MR3726904

2017
[19]

Cambridge Series in Statistical and Probabilistic Mathematics, 42

Lyons, R., and Peres, Y., Probability on trees and networks . Cambridge Series in Statistical and Probabilistic Mathematics, 42. Cambridge University Press, New York, 2016. MR3616205

2016
[21]

Electron

Oliveira, R., Mixing and hitting times for ﬁnite Markov chains. Electron. J. Probab. 17 (2012), no. 70, 12 pp. MR2968677

2012
[22]

Oliveira, R., On the coalescence time of reversible random walks. Trans. Amer. Math. Soc. 364 (2012), no. 4, 2109–2128. MR2869200

2012
[23]

Peres, Y., and Sousi, P., Mixing times are hitting times of large sets . J. Theoret. Probab. 28 (2015), no. 2, 488–519. MR3370663

2015
[24]

Electron

Peres, Y., Sauerwald, T., Sousi, P., and Stauﬀer, A., Intersect ion and mixing times for reversible chains. Electron. J. Probab. 22 (2017), No. 12, 16 pp. MR3613705

2017
[25]

(20 21) arXiv 2102.05597 27

Salez, J., Cutoﬀ for non-negatively curved Markov chains. (20 21) arXiv 2102.05597 27

arXiv

Pith tools

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

[1] [20]

Oliveira, R., Mean ﬁeld conditions for coalescing random walks. Ann. Probab. 41 (2013), no. 5, 3420–3461. MR3127887

work page 2013

[2] [1]

Aldous, D., Hitting times for random walks on vertex-transitive gr aphs. Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 1, 179–191. MR0994089

work page 1989

[3] [2]

Aldous, D., Mixing times and hitting times. (2010). Available at http://www.stat.berkeley. edu/ aldous/Talks/slides.html

work page 2010

[4] [3]

Journal of the London Mathematical Society, 2(3):564–576, 1982

Aldous, D., Some inequalities for reversible Markov chains. Journal of the London Mathematical Society, 2(3):564–576, 1982. MR657512

work page 1982

[5] [4]

Aldous, D., Threshold limits for cover times. J. Theoret. Probab. 4 (1991), no. 1, 197–211. MR1088401

work page 1991

[6] [5]

Unﬁnished manuscript

Aldous, D., and Fill, J., Reversible Markov chains and random walks on graphs . Unﬁnished manuscript. Available at the ﬁrst author’s website

work page

[7] [6]

Basu, R., Hermon, J., and Peres, Y., Characterization of cutoﬀ f or reversible Markov chains. Ann. Probab. 45 (2017), no. 3, 1448–1487. MR3650406

work page 2017

[8] [7]

Y., and Saloﬀ-Coste, L., The cutoﬀ phenomenon for erg odic Markov processes

Chen, G. Y., and Saloﬀ-Coste, L., The cutoﬀ phenomenon for erg odic Markov processes. Electron. J. Probab. 13 (2008), no. 3, 26–78. MR2375599

work page 2008