Mixing Phases and Metastability for the Glauber Dynamics on the p-Spin Curie-Weiss Model

Jiang Zhang; Ramkrishna Jyoti Samanta; Somabha Mukherjee

REVIEW 2 major objections 4 minor 1 cited by

Mixing Phases and Metastability for the Glauber Dynamics on the p-Spin Curie-Weiss Model

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For the p-spin Curie-Weiss model with $p \geq 3$, the mixing time of the Glauber dynamics takes exactly three orders of magnitude — $\Theta(N \log N)$, $\Theta(N^{3/2})$, or $\exp(\Omega(N))$ — depending on the local maximizers of a…

desk verdict Solid new phase diagram for p-spin Curie-Weiss Glauber dynamics, but the S_p upper bound needs a missing diffusion estimate spelled out. read the letter →

arxiv 2412.16952 v2 pith:U2OZTGP2 submitted 2024-12-22 math.PR math.STstat.TH

Ramkrishna Jyoti Samanta , Somabha Mukherjee , Jiang Zhang This is my paper · ORCID

classification math.PRmath.STstat.TH MSC 60K3560J10

keywords GlauberdynamicsmixingtimeCurie-Weissmodelp-spinmetastabilityphasetransitionmean-fieldIsingMarkovchainMonteCarlo

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

This paper proves that the speed at which the heat-bath Glauber dynamics converges to equilibrium in the $p$-spin Curie-Weiss model is controlled entirely by the local maximizers of a one-dimensional free-energy function $H_{\beta,h,p}(x)=\beta x^p+hx-I(x)$, where $I$ is the binary entropy. For $p \geq 3$, it identifies three regions of the $(\beta,h)$ parameter space: where $H$ has a unique stable maximizer, mixing takes $\Theta(N \log N)$; where $H$ has multiple local maximizers, mixing takes at least $\exp(\Omega(N))$; and where the unique maximizer has zero curvature, mixing takes $\Theta(N^{3/2})$. Almost the whole parameter plane is covered, with only a curve of stationary inflection points left open. The paper also constructs a restricted Glauber dynamics that, even in the metastable multi-maximizer region, mixes in $\Theta(N \log N)$, allowing approximate simulation in polynomial time.

What carries the argument

The central object is the mean-field free-energy function $H_{\beta,h,p}(x)=\beta x^p+hx-I(x)$ on $[-1,1]$, together with the mean-magnetization update map $\lambda(c)=\tanh(p\beta c^{p-1}+h)$; fixed points of $\lambda$ are exactly stationary points of $H$, and the sign of $\lambda'(c_*)-1$ equals the sign of $H''(c_*)$. The argument runs through the drift identity $\mathbb{E}(c_{t+1}-c_t \mid c_t=c)=(\lambda(c)-c)/N$, burn-in epochs with exponential-in-$\sqrt{N}$ concentration, a coupling that contracts expected Hamming distance when $\lambda'$ is bounded below $1$ near the unique fixed point, a bottleneck-ratio bound for the exponential lower bound, and a Taylor expansion at the zero-curvature fixed point for the $N^{3/2}$ critical law. The restricted dynamics in the metastable region is analyzed as a birth-and-death chain on the sum $S_t^+$ with rejection at the basin boundary, controlled through expected hitting times.

What would settle it

Take a point on the boundary curve $B_p$, for example for $p=4$ the curve $h=U(\beta)$ where $H$ has a stationary inflection point alongside a local maximizer, simulate the Glauber dynamics for increasing $N$, and estimate $t_{\mathrm{mix}}(0.35)$; the paper guarantees only $\Omega(N^{4/3})$ there, so observing $\Theta(N\log N)$ or $\exp(\Omega(N))$ would show the three-regime classification misses a fourth regime, while observing $\Theta(N^{3/2})$ would support the natural extrapolation.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.1: for every $\varepsilon \in (0,1/2)$, $p \geq 3$ and $(\beta,h) \in \Theta$, if the point is $p$-locally regular then $t_{\mathrm{mix}}(\varepsilon)=\Theta_\varepsilon(N \log N)$; if it is $p$-locally critical then $t_{\mathrm{mix}}(\varepsilon) \geq \exp(\Omega_\varepsilon(N))$; and if it is $p$-special then $t_{\mathrm{mix}}(\varepsilon)=\Theta_\varepsilon(N^{3/2})$. The regions are defined by the number of local maximizers of $H_{\beta,h,p}$ and the sign of $H''$ at the maximizer, and a direct corollary is that non-global local maximizers — metastable states — are what cause exponentially slow mixing, not the global maximizer structure alone. The complement of the three regions is a one-dimensional curve on which $H$ has a stationary inflection point together with a local maximizer; the paper leaves the exact mixing order there open and proves only a crude polynomial lower bound $\Omega(N^{4/3})$. Theorem 2.2 adds that, in the metastable region, a restricted chain that rejects moves crossing the basin boundary of the largest local maximizer mixes in $\Theta(N \log N)$ to its stationary distribution.

Load-bearing premise

The whole three-regime phase diagram rests on previously established facts about the stationary magnetization — its fluctuations are of order $N^{-1/2}$ in the regular and critical regions, its scaling is $N^{-1/4}$ at special points, and the second-derivative shape of $H$ is exactly as assumed for every $p \geq 3$; if any of those facts fails at some parameter values, the claimed exponents and phase boundaries would need revision.

Editorial extensions

If this is right

Away from the boundary curve, the computational cost of exact simulation from the $p$-spin Curie-Weiss model is now known to three orders: fast $N \log N$, critical $N^{3/2}$, and exponentially slow.
In the metastable region, starting the chain near a non-global local maximizer causes an exponentially long delay, so the practical prescription is to initialize inside the concave basin of a global maximizer.
For $p \geq 3$ and $h=0$, the mixing transition threshold $\beta'_p$ is strictly smaller than the threshold $\tilde{\beta}_p$ for the asymptotics of the magnetization, so there is a window where magnetization looks Gaussian but mixing is already exponentially slow.
The restricted dynamics of Theorem 2.2 lets one approximate samples from a $p$-locally critical model in $O(N \log N)$ time by running parallel restricted chains around each global maximizer and mixing their outputs with Gaussian weights.
The boundary curve $B_p$ is a genuine gap in the classification: the paper proves only $t_{\mathrm{mix}}=\Omega(N^{4/3})$ there, and the exact order remains open.

Reading between the lines

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

[Editorial inference] The boundary curve likely carries its own mixing exponent, plausibly $N^{4/3}$ up to logarithmic factors, because the crude lower bound saturates a Taylor expansion around the stationary inflection point; this extrapolation is not claimed in the paper.
[Editorial inference] The criterion 'mixing is fast exactly when the free-energy function has one stable local maximizer' should transfer to other mean-field spin systems such as the $p$-spin Curie-Weiss Potts model, where the paper names a counting question as the natural next step.
[Editorial inference] The restricted-dynamics construction yields a concrete sampling recipe — run parallel chains in each basin for $C N \log N$ steps and combine them with weights proportional to $(m_i^2-1)H''(m_i)^{-1/2}$ — that could be benchmarked numerically against naive Glauber dynamics in the metastable region.

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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

Bottom line: this is the first essentially complete mixing-time phase diagram for p-spin Curie-Weiss with external field, and the restricted fast sampler for metastable states is a genuine addition. The p=2 trichotomy is prior work, but the p≥3 geometry, with slow mixing starting before the magnetization phase transition, is new. The proof structure is credible: drift and coupling for R_p, a bottleneck lower bound for C_p, and a convex-minorant drift argument for S_p. The dependence on Mukherjee-Son-Bhattacharya (2021) is real but not circular; those theorems are about stationary fluctuations and phase geometry, not mixing times, so this is a reliance on prior results, not a circular derivation.

The softest spot is the upper-bound half of the S_p case. The proof derives θ_{t+1} ≤ θ_t − (c/N)θ_t^3 and then jumps to the conclusion that τ0 = min{t: |e_t| ≤ 1/N} satisfies P(τ0 > aN^{3/2}) → 0, citing Step 1 of Theorem 4.1 in LLP. That deterministic inequality only gets |e| down to order N^{−1/4} in N^{3/2} steps; reaching 1/N needs a separate diffusion or absorption estimate that is not written out. The LLP analogy may supply it, but as it stands the O(N^{3/2}) upper bound is not fully demonstrated. This is an omitted argument, not a known counterexample, but it is load-bearing.

Two smaller issues. The abstract and Theorem 2.1 state exp(Ω(N)) for C_p while the introduction says e^{Ω(√N)}; those should be consistent. And the claim that the parallel-restricted sampler produces 'very nearly a sample' from the Curie-Weiss measure is too loose; either quantify the total variation distance or soften the phrase.

The boundary curve Bp is left open, which is an honest limitation. The crude N^{4/3} lower bound there is fine as a start. For a reader working on mean-field Glauber dynamics or metastability, this paper is worth the time. I would send it to a serious referee, with the request to either supply the missing diffusion estimate in the S_p upper bound or adjust the statement. If that step is fixed, the paper is a solid contribution.

Referee Report

2 major / 4 minor

Summary. The paper studies heat-bath Glauber dynamics for the p-spin Curie-Weiss model with external field h, for p ≥ 3. It partitions the parameter space into three sets: R_p, where H_{β,h,p} has a unique local maximizer with negative second derivative and no other stationary point; C_p, where H has multiple local maximizers; and S_p, where H has a unique local maximizer with zero second derivative. Theorem 2.1 asserts mixing times Θ(N log N), e^{Ω(N)}, and Θ(N^{3/2}) in these three regions, leaving only a one-dimensional boundary curve B_p unresolved. Theorem 2.2 constructs a restricted Glauber dynamics in the metastable case that mixes in time Θ(N log N). The proofs use drift and burn-in estimates, a path-coupling contraction argument, a bottleneck lower bound, and a convex-minorant drift analysis, leaning on fluctuation results from Mukherjee, Son and Bhattacharya (2021).

Significance. If the main theorem is fully established, the paper gives an almost complete scaling classification of Glauber mixing times for the p-spin Curie-Weiss model, extending the classical results of Levin, Luczak and Peres (2007) from p = 2 to p ≥ 3 in the presence of a field. The explicit phase geometry in Appendix A, the matching lower bounds, and the restricted-dynamics mixing result in Theorem 2.2 are substantive contributions. The proofs are structured and mostly credible, and the paper makes good use of prior rigorous results on magnetization fluctuations. The main weaknesses are two upper-bound steps that are delegated to sketched analogies with existing proofs rather than written out; these steps are load-bearing for the central theorem.

major comments (2)

[Section 3.3, Eq. (3.12)] The proof of the S_p upper bound jumps from the cubic drift inequality θ_{t+1} ≤ θ_t − (c/N)θ_t^3 to the conclusion that P(τ0 > aN^{3/2}) → 0 uniformly in N, where τ0 = min{t : |e_t| ≤ 1/N}. This implication is not justified by the displayed inequality alone: the deterministic flow dθ/dt = −(c/N)θ^3 starting from θ = O(N^{−1/4}) reaches order N^{−1} only on a time scale of order N^3, not N^{3/2}. The O(N^{3/2}) hitting-time claim therefore depends on the diffusive fluctuations of the magnetization chain, and requires a separate submartingale or second-moment estimate for the process e_t. Moreover, the target scale 1/N is far below the N^{−1/4} stationary fluctuation scale given by Mukherjee-Son-Bhattacharya Theorem 3.1(3), so the analogy with Theorem 4.1 of Levin-Luczak-Peres (2007) is not immediate. This missing estimate is needed before the upper bound in Theorem 2.1(3) is rigorous.
[Section 3.1, proof for λ′(c∗) ≤ −1] In the p-locally regular case, the paper handles the case λ′(c∗) ≤ −1 by first showing that e_t hits the small neighborhood |e_t| ≤ 1/N in time O(N log N), and then saying that 'the same arguments' as in the proof of Theorem 4.1 of Levin-Luczak-Peres (2007) complete the upper bound. This is not automatic: Lemma 3.4 requires sup_c |λ′(c)| < 1, and the burn-in reduction only guarantees contraction inside a neighborhood of c∗, not a global path-coupling contraction when |λ′(c∗)| > 1. Theorem 4.1 of LLP concerns the case λ′(0) = 2β ∈ (−1,1), so the cited argument does not by itself cover the oscillatory drift case. The manuscript should either provide a coupling or contraction argument that works when |λ′(c∗)| > 1 after the hitting time τ0, or identify precisely which part of LLP applies and why.

minor comments (4)

[Introduction, Section 1] The introduction states that the mixing time for p-locally critical points is e^{Ω(√N)}, while Theorem 2.1 and the abstract state e^{Ω(N)}; this should be corrected to e^{Ω(N)}.
[Throughout] There are several typos, such as 'Theoorem' in Section 3 and 'exitence' in Section 3.3, which should be fixed in a revision.
[Figure 2] The simulation figure would be more informative if the axes and the precise stopping rule for the capped mixing times were described, and if the number of repetitions or error bars were reported.
[Appendix D] The proof of Proposition D.1 cites an unnamed displayed inequality after the definition of A_t; the displayed inequality should be numbered or referenced explicitly for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three mixing-time regimes are proved from model dynamics and prior published phase-geometry/fluctuation results, not from the target theorem.

full rationale

The derivation chain is not circular. Theorem 2.1's regimes R_p, C_p, and S_p are defined by stationary-point geometry of H_{β,h,p}—the number of local maximizers and the sign of H''—and the proof supplies drift, coupling, bottleneck, and hitting-time arguments to derive the corresponding mixing-time exponents. No equation in the paper defines a regime in terms of the mixing time itself, and no fitted parameter is subsequently renamed as a prediction; the bounds are stated directly in terms of the fixed model parameters β, h, and p. The citations to Mukherjee-Son-Bhattacharya (2021) are frequent and load-bearing for magnetization fluctuation widths (Theorem 3.1) and phase geometry (Lemmas F.2 and F.3), but these are prior published theorems about the stationary measure and the function H, not about mixing times, and their assumptions do not include the conclusions of Theorem 2.1; the overlap of one coauthor does not make them circular. The reliance on Levin-Luczak-Peres (2007) is methodological (path coupling, birth-death estimates, hitting-time martingales), not an import of the p-spin result. The one genuine soft spot, also acknowledged in the manuscript's discussion of the remaining boundary curve, is the S_p upper-bound argument in Section 3.3: inequality (3.12) gives a cubic decay of θ_t, but the jump to P(τ0 > aN^{3/2}) → 0 is asserted by analogy with Step 1 of Theorem 4.1 of Levin-Luczak-Peres without writing out the needed diffusion/absorption estimate. This is an omitted proof or correctness risk, not circularity, because nothing in the argument assumes the O(N^{3/2}) conclusion. Accordingly, the paper is self-contained against external benchmarks and has no fitted-input or definitional circularity.

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

The central theorem introduces no fitted constants. The main external debts are to Markov chain mixing theory (LLP 2006) and to Mukherjee-Son-Bhattacharya (2021) for stationary fluctuation and phase-geometry results; these are cited and used as inputs, not as circular restatements of the mixing-time conclusion.

assumptions (4)

standard math Standard Markov chain and coupling tools: bottleneck ratio theorem (Levin-Peres-Wilmer 2006, Theorem 7.3), Chernoff bounds, and hitting-time arguments.
Used throughout Sections 3.1-3.3 and Appendix B to convert drift and bottleneck estimates into mixing-time bounds.
domain assumption Magnetization fluctuation theorems from Mukherjee, Son and Bhattacharya (2021), Theorem 3.1: in R_p, Xbar - c* = O_P(N^{-1/2}); at S_p, N^{1/4}(Xbar - c*) converges weakly.
Used in lower-bound proofs in Sections 3.1 and 3.3 and in Appendix D to ensure the stationary measure concentrates around the relevant maximizer.
domain assumption Phase geometry of H from Mukherjee, Son and Bhattacharya (2021), Lemmas F.2 and F.3: explicit p-special points and root structure of H'' for all p >= 3.
Used in Lemma A.1 and Lemma C.2 to partition the parameter plane into R_p, C_p, S_p and B_p.
domain assumption Large-deviation estimate for magnetization from the proof of Lemma 3.7 in Mukherjee, Son and Bhattacharya (2021), giving an exp(N(H(m1+epsilon/2)-H(m1))) O(N^{3/2}) upper bound.
Used in the bottleneck estimate in Section 3.2 to show the escape probability from a metastable set is exponentially small.

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Pith. "Pith review of Mixing Phases and Metastability for the Glauber Dynamics on the p-Spin Curie-Weiss Model." pith.science (2026). https://pith.science/paper/U2OZTGP2

@misc{pith2026241216952,
  author       = {Pith},
  title        = {Pith review of: Mixing Phases and Metastability for the Glauber Dynamics on the p-Spin Curie-Weiss Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2OZTGP2}},
  note         = {Machine review of arXiv:2412.16952}
}

abstract

The Glauber dynamics for the classical $2$-spin Curie-Weiss model on $N$ nodes with inverse temperature $\beta$ and zero external field is known to mix in time $\Theta(N\log N)$ for $\beta < \frac{1}{2}$, in time $\Theta(N^{3/2})$ at $\beta = \frac{1}{2}$, and in time $\exp(\Omega(N))$ for $\beta >\frac{1}{2}$. In this paper, we consider the $p$-spin generalization of the Curie-Weiss model with an external field $h$, and identify three disjoint regions almost exhausting the parameter space, with the corresponding Glauber dynamics exhibiting three different orders of mixing times in these regions. The construction of these disjoint regions depends on the number of local maximizers of a certain function $H_{\beta,h,p}$, and the behavior of the second derivative of $H_{\beta,h,p}$ at such a local maximizer. Specifically, we show that if $H_{\beta,h,p}$ has a unique local maximizer $m_*$ with $H_{\beta,h,p}''(m_*) < 0$ and no other stationary point, then the Glauber dynamics mixes in time $\Theta(N\log N)$, and if $H_{\beta,h,p}$ has multiple local maximizers, then the mixing time is $\exp(\Omega(N))$. Finally, if $H_{\beta,h,p}$ has a unique local maximizer $m_*$ with $H_{\beta,h,p}''(m_*) = 0$, then the mixing time is $\Theta(N^{3/2})$. We provide an explicit description of the geometry of these three different phases in the parameter space, and observe that the only portion of the parameter plane that is left out by the union of these three regions, is a one-dimensional curve, on which the function $H_{\beta,h,p}$ has a stationary inflection point. Finding out the exact order of the mixing time on this curve remains an open question. Finally, we show that if $H_{\beta,h,p}$ has multiple local maximizers (metastable states), then one can create a restricted version of the original Glauber dynamics, which still mixes in time $\Theta(N\log N)$.

Figures

Figures reproduced from arXiv: 2412.16952 by the authors.

**Figure 2.** Plot of the mixing times (capped at 10,000) of the Glauber dynamics against N for the 4-locally regular point (0.054, 0.5), the 4-special point (1/3, 0.41) and the 4-locally critical point (0.51, 0.184), compared against their theoretical estimates. coincide at 1 2 , while for p ≥ 3 and h = 0, it will follow from Lemma A.1 in Appendix A that the mixing time transition threshold β ′ p is strictly smaller than the tr… view at source ↗

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Works this paper leans on

12 extracted references · 11 canonical work pages · cited by 1 Pith paper

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Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime

Aizenman, M.and Holley, R.(1987). Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime. In Spatial Stochastic Processes, (R. Durrett and H. Kesten, eds.). IMA Volumes in Mathematics and its Applications 8 1–11. Springer, New York. Aslanov, L. A.(1988). Crystal symmetry and atomic interactions.Computers & Mathematics...

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[2]

1 − 2 p p−4 2 < 0 . Finally, note that for odd p, Lemma F.2 (1) in Mukherjee, Son and Bhattacharya (2021) says that c∗ is the unique stationary point of H, and hence by Lemma C.1, the unique fixed point of λ. If p is even and h > 0, then once again, it follows from the proof of Lemma F.2 (2) in Mukherjee, Son and Bhattacharya (2021) that H ′(x) > 0 on [0 ...

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p p − 2 p−2 2 , ˆhp = tanh−1 rp − 2 p − ˆβpp p − 2 p p−1 2 . /Mixing of the Glauber Dynamics on thep-Spin Curie-Weiss Model 24 Also, define the following two threshold parameters, which are important quantities in describing the phase geometry: ˜βp := sup ( β >0 : sup x∈[0,1] Hβ,0,p(x) = 0 ) = inf x∈[0,1] I(x) xp , β′ p := sup ( β >0 : sup x∈[0,1] H ′ β,0...

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This proves (d)

Further, βpa1(β)p−1 = Op β1− p−1 p−2 = Op β− 1 p−2 . This proves (d). Finally, note that as β → ˆβ+ p , both the roots a1(β) and a2(β) of H ′′ β,0,p also converge to the only root q 1 − 2 p of H ′′ ˆβp,0,p. Hence, both U (β) and L(β) converge to ˆhp. This completes the proof of part (1) of Lemma A.1. /Mixing of the Glauber Dynamics on thep-Spin Curie-Weis...

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On the other hand, we can choose β >0 large enough, such that H ′ β,0,p( 1 2 ) > tanh−1(wp) > 0, which would then imply that H ′ β,0,p(a2(β)) > tanh−1(wp)

Next, since a1 < wp := q 1 − 2 p , we have H ′ β,0,p(a1(β)) ≥ −tanh−1(wp). On the other hand, we can choose β >0 large enough, such that H ′ β,0,p( 1 2 ) > tanh−1(wp) > 0, which would then imply that H ′ β,0,p(a2(β)) > tanh−1(wp). So, g(β) > 0 for all large β. Hence, there is a unique root β0 of g, larger than β′ p, and for β ∈ ( ˆβp, β0), U is strictly d...

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By Lemma F.2 in Mukherjee, Son and Bhattacharya (2021), the only p-special point for odd p is ( ˆβp, ˆhp), and the only p-special points for even p are ( ˆβp, ±ˆhp)

p p − 2 p−2 2 , ˆhp = tanh−1 rp − 2 p − ˆβpp p − 2 p p−1 2 . By Lemma F.2 in Mukherjee, Son and Bhattacharya (2021), the only p-special point for odd p is ( ˆβp, ˆhp), and the only p-special points for even p are ( ˆβp, ±ˆhp). In this case, the function H = Hβ,h,p has a unique global maximizer c∗ := i p 1 − 2/p, if the p-special point under consideration ...

work page 2021
[10]

(1) If H ′ β,0,p(a2) ≤ 0, then Hβ,h,p has more than one local maximizer if and only if −H ′ β,0,p(a2) < h <−H ′ β,0,p(a1)

Then, the following are true. (1) If H ′ β,0,p(a2) ≤ 0, then Hβ,h,p has more than one local maximizer if and only if −H ′ β,0,p(a2) < h <−H ′ β,0,p(a1). (2) If H ′ β,0,p(a2) > 0, then Hβ,h,p has more than one local maximizer if and only if h < − min{H ′ β,0,p(a1), H′ β,0,p(−a2)}. Proof. We highlight on the following monotonicity pattern ofH ′ β,h,p (see t...

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Moreover, once again, H ′ β,h,p(a2) > 0, which guarantees another local maximizer in ( a2, 1)

So, H ′ β,h,p has a root in ( −1, −a2), which is a local maximizer of Hβ,h,p. Moreover, once again, H ′ β,h,p(a2) > 0, which guarantees another local maximizer in ( a2, 1). This proves the if part, and completes the proof of Lemma C.3. Appendix D: A Crude Lower Bound of the Mixing Time onBp In this section, we show how the arguments in the proof of Theore...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime

Aizenman, M.and Holley, R.(1987). Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime. In Spatial Stochastic Processes, (R. Durrett and H. Kesten, eds.). IMA Volumes in Mathematics and its Applications 8 1–11. Springer, New York. Aslanov, L. A.(1988). Crystal symmetry and atomic interactions.Computers & Mathematics...

arXiv 1987

[2] [2]

1 − 2 p p−4 2 < 0 . Finally, note that for odd p, Lemma F.2 (1) in Mukherjee, Son and Bhattacharya (2021) says that c∗ is the unique stationary point of H, and hence by Lemma C.1, the unique fixed point of λ. If p is even and h > 0, then once again, it follows from the proof of Lemma F.2 (2) in Mukherjee, Son and Bhattacharya (2021) that H ′(x) > 0 on [0 ...

work page 2021

[3] [5]

p p − 2 p−2 2 , ˆhp = tanh−1 rp − 2 p − ˆβpp p − 2 p p−1 2 . /Mixing of the Glauber Dynamics on thep-Spin Curie-Weiss Model 24 Also, define the following two threshold parameters, which are important quantities in describing the phase geometry: ˜βp := sup ( β >0 : sup x∈[0,1] Hβ,0,p(x) = 0 ) = inf x∈[0,1] I(x) xp , β′ p := sup ( β >0 : sup x∈[0,1] H ′ β,0...

work page 2021

[4] [6]

This proves (d)

Further, βpa1(β)p−1 = Op β1− p−1 p−2 = Op β− 1 p−2 . This proves (d). Finally, note that as β → ˆβ+ p , both the roots a1(β) and a2(β) of H ′′ β,0,p also converge to the only root q 1 − 2 p of H ′′ ˆβp,0,p. Hence, both U (β) and L(β) converge to ˆhp. This completes the proof of part (1) of Lemma A.1. /Mixing of the Glauber Dynamics on thep-Spin Curie-Weis...

work page 2021

[5] [7]

On the other hand, we can choose β >0 large enough, such that H ′ β,0,p( 1 2 ) > tanh−1(wp) > 0, which would then imply that H ′ β,0,p(a2(β)) > tanh−1(wp)

Next, since a1 < wp := q 1 − 2 p , we have H ′ β,0,p(a1(β)) ≥ −tanh−1(wp). On the other hand, we can choose β >0 large enough, such that H ′ β,0,p( 1 2 ) > tanh−1(wp) > 0, which would then imply that H ′ β,0,p(a2(β)) > tanh−1(wp). So, g(β) > 0 for all large β. Hence, there is a unique root β0 of g, larger than β′ p, and for β ∈ ( ˆβp, β0), U is strictly d...

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[6] [8]

By Lemma F.2 in Mukherjee, Son and Bhattacharya (2021), the only p-special point for odd p is ( ˆβp, ˆhp), and the only p-special points for even p are ( ˆβp, ±ˆhp)

p p − 2 p−2 2 , ˆhp = tanh−1 rp − 2 p − ˆβpp p − 2 p p−1 2 . By Lemma F.2 in Mukherjee, Son and Bhattacharya (2021), the only p-special point for odd p is ( ˆβp, ˆhp), and the only p-special points for even p are ( ˆβp, ±ˆhp). In this case, the function H = Hβ,h,p has a unique global maximizer c∗ := i p 1 − 2/p, if the p-special point under consideration ...

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[7] [10]

(1) If H ′ β,0,p(a2) ≤ 0, then Hβ,h,p has more than one local maximizer if and only if −H ′ β,0,p(a2) < h <−H ′ β,0,p(a1)

Then, the following are true. (1) If H ′ β,0,p(a2) ≤ 0, then Hβ,h,p has more than one local maximizer if and only if −H ′ β,0,p(a2) < h <−H ′ β,0,p(a1). (2) If H ′ β,0,p(a2) > 0, then Hβ,h,p has more than one local maximizer if and only if h < − min{H ′ β,0,p(a1), H′ β,0,p(−a2)}. Proof. We highlight on the following monotonicity pattern ofH ′ β,h,p (see t...

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[8] [11]

Moreover, once again, H ′ β,h,p(a2) > 0, which guarantees another local maximizer in ( a2, 1)

So, H ′ β,h,p has a root in ( −1, −a2), which is a local maximizer of Hβ,h,p. Moreover, once again, H ′ β,h,p(a2) > 0, which guarantees another local maximizer in ( a2, 1). This proves the if part, and completes the proof of Lemma C.3. Appendix D: A Crude Lower Bound of the Mixing Time onBp In this section, we show how the arguments in the proof of Theore...

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