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On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model

Abstract

We consider the coupling from the past implementation of the random–cluster heat-bath process, and study its random running time, or coupling time. We focus on hypercubic lattices embedded on tori, in dimensions one to three, with cluster fugacity at least one. We make a number of conjectures regarding the asymptotic behaviour of the coupling time, motivated by rigorous results in one dimension and Monte Carlo simulations in dimensions two and three. Amongst our findings, we observe that, for generic parameter values, the distribution of the appropriately standardized coupling time converges to a Gumbel distribution, and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time. Perhaps surprisingly, we observe these results to hold both off criticality, where the coupling time closely mimics the coupon collector’s problem, and also at the critical point, provided the cluster fugacity is below the value at which the transition becomes discontinuous. Finally, we consider analogous questions for the single-spin Ising heat-bath process.

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Notes

  1. For concreteness, in the present discussion we refer to the measure corresponding to wired boundary conditions [24, Sect. 4.2].

  2. We adopt the convention that \(\mathbb {N}\,:=\{0,1,2,\ldots \}\) and \(\mathbb {N}^+:=\{1,2,\ldots \}\).

  3. The notation \(a_L\asymp b_L\) means that there exist constants \(c,C>0\) such that \(c b_L \le a_L \le C b_L\) for all sufficiently large L.

  4. Assuming the relevant exponents exist.

  5. I.e. \(M_n\mapsto (M_n-b_n)/a_n\) for some deterministic sequences \(a_n>0\) and \(b_n\).

  6. Since the time \(W_k\) of the first arrival of \(D_k\) is not geometrically distributed, \(I_k\) is not itself a sum of geometric random variables.

References

  1. Aizenman, M., Duminil-Copin, H., Sidoravicius, V.: Random currents and continuity of ising model’s spontaneous magnetization. Commun. Math. Phys. 334, 719–742 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. Baxter, R.J.: Solvable eight-vertex model on an arbitrary planar lattice. Philos. Trans. R. Soc. A 289, 315–346 (1978)

    ADS  MathSciNet  Article  Google Scholar 

  3. Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random-cluster model is critical for q \(\ge \) 1. Probab. Theory Relat. Fields 153, 511–542 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. Billingsley, P.: Probability and Measure. (Wiley Series in Probability and Statistics), 3rd edn. Wiley, New York (1994)

    MATH  Google Scholar 

  5. Cesi, F., Guadagni, G., Martinelli, F., Schonmann, R.H.: On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point. J. Stat. Phys. 85, 55–102 (1996)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. Chayes, L., Machta, J.: Graphical representations and cluster algorithms II. Phys. A 254, 477–516 (1998)

    Article  Google Scholar 

  7. Chow, Y.S., Teicher, H.: Probability Theory: Independence, Interchangeability, Martingales. Springer, New York (1978)

    Book  MATH  Google Scholar 

  8. Deng, Y., Blöte, H.: Simultaneous analysis of several models in the three-dimensional Ising universality class. Phys. Rev. E 68, 036125 (2003)

    ADS  Article  Google Scholar 

  9. Deng, Y., Garoni, T., Machta, J., Ossola, G., Polin, M., Sokal, A.: Critical behavior of the Chayes–Machta–Swendsen–Wang dynamics. Phys. Rev. Lett. 99, 055701 (2007)

    ADS  Article  Google Scholar 

  10. Deng, Y., Garoni, T.M., Sokal, A.D.: Critical speeding-up in the local dynamics of the random-cluster model. Phys. Rev. Lett. 98, 230602 (2007)

    ADS  Article  Google Scholar 

  11. Duminil-Copin, H., Gagnebin, M., Harel, M., Manolescu, I., Tassion, V.: Discontinuity of the phase transition for the planar random-cluster and Potts models with \(q > 4\). arXiv:1611.09877 (2016)

  12. Duminil-Copin, H., Sidoravicius, V., Tassion, V.: Continuity of the phase transition for planar random-cluster and Potts models with \(1 \le q \le 4\). arXiv:1505.04159 (2015)

  13. Dyer, M., Greenhill, C., Ullrich, M.: Structure and eigenvalues of heat-bath Markov chains. Linear Algebra Appl. 454, 57–71 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  14. Elci, E.: Algorithmic and geometric aspects of the random-cluster model. Ph.D. thesis (2015)

  15. Elçi, E.M., Weigel, M.: Efficient simulation of the random-cluster model. Phys. Rev. E 88, 033303 (2013)

    ADS  Article  Google Scholar 

  16. Elçi, E.M., Weigel, M.: Dynamic connectivity algorithms for Monte Carlo simulations of the random-cluster model. J. Phys. 510(1), 012013 (2014)

    Google Scholar 

  17. Erdos, P., Renyi, A.: On a classical problem of probability theory. Publ. Math. Inst. Hung. Acad. Sci. Ser. A 6, 215–219 (1961)

    MathSciNet  MATH  Google Scholar 

  18. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)

    MATH  Google Scholar 

  19. Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems. A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2016)

    MATH  Google Scholar 

  20. Gheissari, R., Lubetzky, E.: Mixing Times Of Critical 2D Potts Models. arXiv:1607.02182 (2016)

  21. Gliozzi, F.: Simulation of Potts models with real q and no critical slowing down. Phys. Rev. E 66, 016115 (2002)

    ADS  Article  Google Scholar 

  22. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. A Foundation for Computer Science, 2nd edn. Addisen-Wesley Publishing Company, Boston (1994)

    MATH  Google Scholar 

  23. Grassberger, P.: Damage spreading and critical exponents for “model A” Ising dynamics. Phys. A 214, 547–559 (1995)

    MathSciNet  Article  Google Scholar 

  24. Grimmett, G.: The Random-Cluster Model. Springer, New York (2006)

    Book  MATH  Google Scholar 

  25. Grimmett, G.: Probability on Graphs. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  26. Guo, H., Jerrum, M.: Random cluster dynamics for the Ising model is rapidly mixing. arXiv:1605.00139 pp. 1–15 (2016)

  27. Häggström, O.: Finite Markov Chains and Algorithmic Applications. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  28. Hartmann, A.: Calculation of partition functions by measuring component distributions. Phys. Rev. Lett. 94, 050601 (2005)

    ADS  Article  Google Scholar 

  29. Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM (JACM) 48, 723–760 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  30. Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Philos. Soc. 108, 35–53 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  31. Janson, S.: Tail bounds for sums of geometric and exponential random variables (2014). http://www2.math.uu.se/~svante/papers/sjN14.pdf

  32. Jerrum, M.: Mathematical Foundations of the Markov Chain Monte Carlo Method. In: Probabilistic Methods for Algorithmic Discrete Mathematics, pp. 116–165. Springer, New York (1998)

  33. Laanait, L., Messager, A., Miracle-Solé, S., Ruiz, J., Shlosman, S.: Interfaces in the Potts model I: Pirogov–Sinai theory of the Fortuin–Kasteleyn representation. Commun. Math. Phys. 140(1), 81–91 (1991)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and related properties of random sequences and processes. Springer, New York (1983)

    Book  MATH  Google Scholar 

  35. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  36. Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhauser, Boston (1996)

    Book  MATH  Google Scholar 

  37. McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  38. Mitzenmacher, M., Upfal, E.: Probability and Computing. Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  39. Nacu, S.: Glauber dynamics on the cycle is monotone. Probab. Theory Relat. Fields 127, 177–185 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  40. Nienhuis, B.: Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys. 34, 731–761 (1984)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. Nightingale, M.P., Bloete, H.W.J.: Dynamic exponent of the two-dimensional Ising model and Monte Carlo computation of the subdominant eigenvalue of the stochastic matrix. Phys. Rev. Lett. 76, 4548–4551 (1996)

    ADS  Article  Google Scholar 

  42. Posfai, A.: Approximation Theorems Related to the Coupon Collector’s Problem . Ph.D. thesis (2010)

  43. Propp, J., Wilson, D.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algorithms 9, 223–252 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  44. Sinclair, A.B., Alistair, Sinclair, A.: Random-Cluster Dynamics in \(\mathbb{Z}^2\). In: Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 498–513 (2016)

  45. Sokal, A.D.: Monte Carlo methods in statistical mechanics: foundations and new algorithms. In: DeWitt-Morette, C., Cartier, P., Folacci, A. (eds.) Functional Integration: Basics and Applications (1996 Cargèse summer school), pp. 131–192. Plenum, New York (1997)

    Chapter  Google Scholar 

  46. Sweeny, M.: Monte Carlo study of weighted percolation clusters relevant to the Potts models. Phys. Rev. B 27, 4445–4455 (1983)

    ADS  Article  Google Scholar 

  47. Swendsen, R.H., Wang, J.S.: Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987)

    ADS  Article  Google Scholar 

  48. Wang, J.S., Kozan, O., Swendsen, R.: Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation. Phys. Rev. E 66(5), 057101 (2002)

    ADS  Article  Google Scholar 

  49. Welsh, D.J.A.: Complexity: Knots, Colourings and Counting, London Mathematical Society Lecture Note Series, vol. 186. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  50. Deng, Youjin, Garoni, Timothy, M., Sokal, Alan, Zhou, Zongzheng: Dynamic critical behavior of the Chayes-Machta random-cluster algorithm II: Three-dimensions. In preparation

  51. Young, P.: Everything You Wanted to Know about Data Analysis and Fitting but were Afraid to ask. Springer, New York (2015)

    Book  Google Scholar 

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Acknowledgements

The authors thank Youjin Deng, Alan Sokal, and Ulli Wolff for useful discussions, and an anonymous referee for helpful comments. This work was supported under the Australian Research Council’s Discovery Projects funding scheme (project numbers DP140100559 & DP110101141), and T.G. is the recipient of an Australian Research Council Future Fellowship (project number FT100100494). A.C. would like to thank STREP project MATHEMACS. The work of EE and MW was partially supported by the European Commission through the IRSES network DIONICOS (PIRSES-GA-2013-612707).

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Appendices

Appendix A: Autocorrelation Functions of Strictly Increasing Observables

Let P denote the transition matrix of the FK heat-bath process on a finite graph \(G=(V,E)\) with parameters \(p\in (0,1)\) and \(q\ge 1\), and let \(k=2^{|E|}\). To avoid trivialities, we assume \(|E|>1\). We regard elements of \(\mathbb {R}^k\) as functions from \(2^E\) to \(\mathbb {R}\), and we endow \(\mathbb {R}^k\) with the inner product \(\langle \cdot ,\cdot \rangle \) defined by

$$\begin{aligned} \langle g,h\rangle := \sum _{A\subseteq E} g(A)\,h(A)\,\phi (A). \end{aligned}$$

Denote the eigenvalues of P by \(1=\lambda _1 > \lambda _2\ge \ldots \ge \lambda _k\). As mentioned in Sect. 2.2, general results for heat-bath chains [13] imply that all \(\lambda _i\) are non-negative. Let \(\{\psi _i\}_{i=1}^k\) be an orthonormal basis for \(\mathbb {R}^k\) such that \(\psi _i\) is an eigenfunction of P corresponding to \(\lambda _i\). The Perron-Frobenius theorem implies that the eigenspace of \(\lambda _1\) is one-dimensional, and that we can take \(\psi _1(A)=1\) for all \(A\subseteq E\). Let W denote the eigenspace of \(\lambda _2\). For \(g\in \mathbb {R}^k\), we let \(g_W\) denote its projection onto W.

We say \(g\in \mathbb {R}^k\) is increasing if \(A\subset B\) implies \(g(A)\le g(B)\), and strictly increasing if \(A\subset B\) implies \(g(A) < g(B)\).

Proposition A.1

Let \((X_t)_{t\in \mathbb {N}}\) be a stationary FK heat-bath process, and for \(g\in \mathbb {R}^k\) define \((g_t)_{t\in \mathbb {N}}\) via \(g_t\,:=g(X_t)\). If g is strictly increasing, then its autocorrelation function satisfies

$$\begin{aligned} \rho _{g}(t) := \frac{{\text {cov}}(g_0,g_t)}{{\text {var}}(g_0)} \sim C e^{-t/t_{\exp }},\qquad t\rightarrow \infty , \end{aligned}$$

for constant \(C>0\).

Proof

Let \(\varPi \) denote the projection matrix onto the space of constant functions. General arguments (see e.g. [45] or [36, Chap. 9]) imply

$$\begin{aligned} {\text {cov}}(g_0,g_t) = \langle g,(P^t-\varPi )g\rangle = \sum _{l=2}^k \langle g,\psi _l\rangle ^2\lambda _l^t = \Vert g_W\Vert ^2\lambda _2^t \,+\, \sum _{l=\dim (W)+2}^k\langle g,\psi _l\rangle ^2\lambda _l^t. \end{aligned}$$

Since g is strictly increasing, Lemma A.2 implies that \(\Vert g_W\Vert ^2>0\), and therefore

$$\begin{aligned} {\text {cov}}(g_0,g_t) \sim \Vert g_W\Vert ^2 e^{-t/t_{\exp }}, \qquad t\rightarrow \infty . \end{aligned}$$

It follows that

$$\begin{aligned} \rho _g(t) \sim \frac{\Vert g_W\Vert ^2}{{\text {var}}(g)} e^{-t/t_{\exp }}, \qquad t\rightarrow \infty . \end{aligned}$$

\(\square \)

Lemma A.2

If g is strictly increasing, then its projection onto W is non-zero.

Proof

Lemma A.3 implies there exists \(\psi \in W\) which is non-zero and increasing. Positive association (see e.g. [24, Theorem 3.8 (b)]) then implies that for any other increasing g we have

$$\begin{aligned} \langle g,\psi \rangle \ge \mathbb {E}(g)\,\mathbb {E}(\psi ) = 0, \end{aligned}$$
(A.1)

since \(\mathbb {E}(\psi )=\langle \psi _1,\psi \rangle =0\). In particular, suppose that g is strictly increasing. Choosing \(\alpha >0\) so that

$$\begin{aligned} g(B)-g(A) > \alpha [\psi (B)-\psi (A)], \qquad \text { for all } A \subset B\subseteq E, \end{aligned}$$

implies that \(g-\alpha \psi \) is also strictly increasing. Applying (A.1) to \(g-\alpha \psi \) then yields

$$\begin{aligned} \langle g-\alpha \psi ,\psi \rangle \ge 0. \end{aligned}$$

Rearranging, and using the fact that \(\psi \) is non-zero then implies

$$\begin{aligned} \langle g,\psi \rangle \ge \alpha \langle \psi ,\psi \rangle >0. \end{aligned}$$

Therefore, g has a non-zero projection onto \(\psi \in W\), and the stated result follows. \(\square \)

The following lemma is the natural analogue, in the FK setting, of the result [39, Lemma 3] established for the Ising heat-bath process.

Lemma A.3

There exists \(\psi \in W\) which is non-zero and increasing.

Proof

Let \(g=\psi _2+ C (\mathscr {N}-\mathbb {E}(\mathscr {N}))\), where \(\mathscr {N}\in \mathbb {R}^k\) is defined so that \(\mathscr {N}(A)=|A|\) for each \(A\subseteq E\), and \(C>0\) is a constant. We have

$$\begin{aligned} g = [1+C\langle \mathscr {N},\psi _2\rangle ]\psi _2 + C \sum _{j=3}^k \langle \mathscr {N},\psi _j\rangle \psi _j. \end{aligned}$$

If \(\langle \mathscr {N},\psi _2\rangle =0\), then g has a non-zero projection onto \(\psi _2\), for any choice of \(C>0\). If \(\langle \mathscr {N},\psi _2\rangle \ne 0\), then choosing \(C>|\langle \mathscr {N},\psi _2\rangle |^{-1}\) suffices to guarantee that g again has a non-zero projection onto \(\psi _2\). In either case, assume C is so chosen. It follows that \(g_W\) is non-zero.

If \(A\subset B\), then

$$\begin{aligned} g(B) - g(A) = \psi _2(B)-\psi _2(A) + C[\mathscr {N}(B)-\mathscr {N}(A)] \ge \min _{A\subset B\subseteq E} [\psi _2(B) - \psi _2(A)] + C. \end{aligned}$$

Therefore, by choosing \(C>\left| \min \limits _{A\subset B\subseteq E} [\psi _2(B) - \psi _2(A)]\right| \) we guarantee that g is increasing. Lemma A.4 then implies that \(g_W\) is increasing. Therefore, \(\psi =g_W\) is an increasing, non-zero element of W. \(\square \)

Lemma A.4

If g is increasing and has zero-mean, then its projection onto W is also increasing.

Proof

Let \(g\in \mathbb {R}^k\) be any increasing observable with mean zero, and let \(t\in \mathbb {N}^+\). Since Lemma A.6 implies \(\lambda _2>0\), we can write

$$\begin{aligned} \frac{P^t g}{\lambda _2^t} = g_W + \sum _{l=\dim (W)+2}^k \langle g,\psi _l\rangle \psi _l\,\left( \frac{\lambda _l}{\lambda _2}\right) ^t. \end{aligned}$$

It follows that

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{P^t g}{\lambda _2^t} = g_W. \end{aligned}$$
(A.2)

Now, for any given \(t\ge 1\), Lemma A.5 implies that \(P^t g(A)\) is an increasing function of A, and so \(\lambda _2^{-t}P^t g (A)\) is also an increasing function of A. It then follows, as an elementary consequence of (A.2), that \(g_W\) is also increasing. We have therefore established that if g is an increasing zero-mean function, then its projection \(g_W\) is also increasing. \(\square \)

Lemma A.5

If \(g\in \mathbb {R}^k\) is increasing, then \(P^tg\) is also increasing, for every \(t\ge 1\).

Proof

Let \((f,\mathscr {E},U)\) be the random mapping representation for P given in Sect. 2.1; see (2.3). Let \(A_1\subset A_2\subseteq E\), and let \(B_i=f(A_i,\mathscr {E},U)\) for \(i=1,2\). Clearly, \((B_1,B_2)\) is a coupling of the distributions \(P(A_1,\cdot )\) and \(P(A_2,\cdot )\), and the monotonicity of f implies \(B_1\subseteq B_2\). Strassen’s theorem (see e.g. [25, Theorem 4.2]) then implies that

$$\begin{aligned} \mathbb {E}_{P(A_1,\cdot )}(g) \le \mathbb {E}_{P(A_2,\cdot )}(g) \end{aligned}$$

for any increasing \(g\in \mathbb {R}^k\). It follows that

$$\begin{aligned} (Pg)(A_1) = \sum _{B\subseteq E} P(A_1,B) g(B)= & {} \mathbb {E}_{P(A_1,\cdot )}(g)\\\le & {} \mathbb {E}_{P(A_2,\cdot )}(g) = \sum _{B\subseteq E} P(A_2,B) g(B) = (Pg)(A_2). \end{aligned}$$

Since this holds for any \(A_1\subset A_2\subseteq E\), it follows that Pg is increasing. It then follows by a simple induction that \(P^tg\) is increasing for any \(t\ge 1\). \(\square \)

Lemma A.6

The second-largest eigenvalue of P is positive.

Proof

Since P is reversible and irreducible we have the spectral decomposition (see e.g. [35, Lemma 12.2])

$$\begin{aligned} \frac{P(A,B)}{\phi (B)} = 1 + \sum _{j=2}^k\psi _i(A)\psi _i(B)\lambda _i. \end{aligned}$$

Since \(\lambda _2\ge \lambda _j\ge 0\) for all \(j>2\), it follows that if \(\lambda _2=0\), then \(P(A,B)=\phi (B)\) for all \(A,B\subseteq E\). But since, by assumption, we have \(|E|>1\), we can choose \(A,B\subseteq E\) with \(|A\triangle B| > 1\), where \(\triangle \) denotes symmetric difference, and (2.2) then implies

$$\begin{aligned} P(A,B) = 0 \ne \phi (B). \end{aligned}$$

We have therefore reached a contradiction, and we conclude that \(\lambda _2>0\). \(\square \)

Appendix B: Coupon Collecting

Let \(n\in \mathbb {N}^+\), and let \(C_1,C_2,\ldots \) be an iid sequence of uniformly random elements of \([n]\,:=\{1,2,\ldots ,n\}\). For \(t\in \mathbb {N}^+\), we think of \(C_t\) as the coupon collected at time t. For \(i\in [n]\), let \(D_i\in [n]\) denote the ith distinct type of coupon collected; i.e. the ith distinct element of the sequence \(C_1,C_2,\ldots \). Let \(S_i(t)\,:=\#\{s\le t : C_s = D_i\}\), the number of copies of \(D_i\) collected by time t. Define \(R_t\,:=\{c\in [n]: C_s = c \text { for some } s\le t\}\), the set of distinct coupon types collected up to time t. For any \(1\le k \le n\), let \(W_k=\inf \{t\in \mathbb {N}^+: |R_t| = k\}\), and note that \(W_k\) is simply the hitting time of \(D_{k}\). The coupon collector’s time is then defined as \(W\,:=W_n\).

For each \(c\in [n]\), define

$$\begin{aligned} H(c) = \sup \{t\le W: C_t=c\}. \end{aligned}$$

We refer to the time \(H(c)\) as the last visit to c. Let \((H_i)_{i=1}^n\) denote the sequence of the \(H(c)\), arranged in increasing order. In particular, \(H_1\) is the first time that a last visit occurs.

Lemma B.1

There exists \(\varphi >0\) such that \(\mathbb {P}(|R_{H_{1}}| \le \left\lfloor \ln n \right\rfloor ) = O(n^{-\varphi })\).

Proof

Inserting \(a_n=\lfloor \ln (n)\rfloor \) and \(c_n=\lfloor \ln (n)/4\rfloor \) into Lemma B.2 and applying the union bound, implies

$$\begin{aligned} \mathbb {P}\left( \bigcup _{i=1}^{a_{n}} \left\{ S_{i}(W) \le c_n \right\} \right)&\le \ln (n) \exp \left( -\frac{1}{2}\ln (n-a_n) + \frac{\ln (n)}{4} + 1\right) \\&= e\,\ln (n) \exp \left( -\frac{1}{4}\ln (n) - \frac{1}{2}\ln \left( 1-\frac{a_n}{n}\right) \right) \\&= \frac{e}{\sqrt{1-\lfloor \ln (n)\rfloor /n}}\,\ln (n)\, n^{-1/4}. \end{aligned}$$

Therefore, for any \(0<\rho <1/4\), we have

$$\begin{aligned} \mathbb {P}\left( \bigcup _{i=1}^{a_{n}} \left\{ S_{i}(W) \le c_n \right\} \right) = O(n^{-\rho }), \qquad n\rightarrow \infty . \end{aligned}$$

It follows that,

$$\begin{aligned} \mathbb {P}(|R_{H_1}| \le a_n) = \mathbb {P}\left( |R_{H_1}|\le a_n, \bigcap _{i=1}^{a_n}\{S_i(W)>c_n\}\right) + O(n^{-\rho }) \end{aligned}$$
(B.1)

Let \(I\,:=\inf \{t\in \mathbb {N}^+: S_i(t) = c_n \text { for some } i\in [n] \}\), the first time that there exists a coupon type for which exactly \(c_n\) copies have been collected, and define the random variable \(K\in [n]\) via \(C_{H_1}=D_K\). If \(|R_{H_1}|\le a_n\), then \(1\le K \le a_n\). Therefore, observing that \(S_{K}(W)=S_{K}(H_1)\), we find

$$\begin{aligned} \mathbb {P}\left( |R_{H_1}|\le a_n, \bigcap _{i=1}^{a_n}\{S_i(W)>c_n\}\right)&\le \mathbb {P}(|R_{H_1}|\le a_n, S_{K}(W)> c_n) \nonumber \\&= \mathbb {P}(|R_{H_1}|\le a_n, S_{K}(H_1) > c_n) \nonumber \\&\le \mathbb {P}(|R_I|\le a_n) \end{aligned}$$
(B.2)

since if \(|R_{H_1}|\le a_n\) and \(S_{K}(H_1) > c_n\) then \(|R_I|\le a_n\). Combining (B.1) and (B.2) then implies

$$\begin{aligned} \mathbb {P}(|R_{H_1}| \le a_n) \le \mathbb {P}(|R_I|\le a_n) + O(n^{-\rho }). \end{aligned}$$

However, Lemma B.3 implies that there exists \(\delta >0\) such that \(\mathbb {P}(|R_I|\le a_n) = O(n^{-\delta })\). We therefore conclude that, if \(\varphi =\min \{\rho ,\delta \}\), then

$$\begin{aligned} \mathbb {P}(|R_{H_{1}}| \le \left\lfloor \ln n \right\rfloor ) = O(n^{-\varphi }). \end{aligned}$$

\(\square \)

Lemma B.2

Let \((a_n)_{n\in \mathbb {N}^+}\) and \((c_n)_{n\in \mathbb {N}^+}\) be any two sequences of natural numbers. For \(n\in \mathbb {N}^+\), if \(a_n<n\) then for each \(1\le i\le a_n\) we have

$$\begin{aligned} \mathbb {P}\left( S_i(W) \le c_n \right) \le \exp \left( -\ln (\sqrt{n-a_n}) + c_n + 1\right) . \end{aligned}$$

Proof

Fix \(n\in \mathbb {N}^+\) and \(1\le i \le a_n\), and assume \(a_n<n\). Adopting the convention \(W_0=0\), for \(0\le k \le n-1\) we define

$$\begin{aligned} Y_i(k) \,:= \sum _{j=W_{k}+1}^{W_{k+1}-1} \mathbf {1}_{\{C_{j}= D_{i}\}}. \end{aligned}$$

Since \(Y_i(k)=0\) for all \(k<i\), and \(C_{W_k}=D_i\) iff \(k=i\), we then have

$$\begin{aligned} S_{i}(W) = 1 + \sum _{k=i}^{n-1} Y_i(k). \end{aligned}$$

And since the random variables \(Y_i(k)\) are independent, for any \(\theta <0\), we have

$$\begin{aligned} \mathbb {P}\left( S_i(W) \le c_n \right)&\le \mathbb {P}\left( \sum _{k=a_n}^{n-1} Y_i(k) \le c_n \right) \nonumber \\&= \mathbb {P}\left( \exp \left[ \theta \sum _{k=a_n}^{n-1} Y_i(k)\right] \ge e^{\theta c_n} \right) \nonumber \\&\le \exp \left( -\theta c_n + \sum _{k=a_n}^{n-1} \ln \mathbb {E}[e^{\theta Y_i(k)}]\right) , \end{aligned}$$
(B.3)

where the final step follows from Markov’s inequality.

The moment generating function of \(Y_i(k)\) can be calculated explicitly. Let \(i \le k \le n-1\). Given \(W_{k}\) and \(W_{k+1}\), the random variable \(Y_i(k)\) has binomial distribution with \(W_{k+1} -W_{k}-1\) trials and success probability 1 / k, which implies

$$\begin{aligned} \mathbb {E}(e^{\theta Y_i(k)})&= \mathbb {E}(\mathbb {E}[e^{\theta Y_i(k)}|W_{k},W_{k+1}]) \\&= \mathbb {E}\left[ \left( \frac{e^{\theta }}{k} + 1 - \frac{1}{k}\right) ^{W_{k+1} - W_{k}-1}\right] . \end{aligned}$$

But since \(W_{k+1}-W_{k}\) has geometric distribution with parameter \(1-k/n\), this becomes

$$\begin{aligned} \mathbb {E}(e^{\theta Y_i(k)}) = \frac{n-k}{n-k+1-e^{\theta }}. \end{aligned}$$

Therefore, setting \(\lambda =1-e^\theta \) and \(b_n=n-a_n\), it follows from the fact that \(\ln (1+\lambda /k)\) is a decreasing function of k that

$$\begin{aligned} -\sum _{k=a_{n}}^{n-1} \ln \mathbb {E}(e^{\theta Y_i(k)})&= \sum _{k=1}^{b_n} \ln \left( 1 + \frac{\lambda }{k} \right) \nonumber \\&\ge \int _1^{b_n} \ln \left( 1+\frac{\lambda }{x}\right) \mathrm {d}\,x \nonumber \\&= \lambda \ln (b_n) + (b_n + \lambda ) \ln (1 + \lambda /b_n) - (1+\lambda )\ln (1+\lambda )\nonumber \\&\ge \lambda \ln (b_n) + \lambda - (1+\lambda )\ln (1+\lambda )\nonumber \\&\ge \lambda \ln (b_n) - 1 \end{aligned}$$
(B.4)

where, in the penultimate step, we used the fact that \(\ln (1+x)\ge x/(1+x)\) holds for all \(x>-1\), and in the last step we used the fact that \((1+\lambda )-(1+\lambda )\ln (1+\lambda )>0\) for any \(\lambda \in (0,1)\). Combining (B.3) and (B.4), we conclude that for all \(\lambda \in (0,1)\) we have

$$\begin{aligned} \mathbb {P}\left( S_i(W) \le c_n \right) \le \exp \left[ -\lambda \ln (n-a_n) -\ln (1-\lambda )c_n + 1) \right] . \end{aligned}$$

Choosing \(\lambda =1/2\) yields the stated result. \(\square \)

Lemma B.3

Fix \(c\in (0,1)\), and define sequences \((a_n)_{n\in \mathbb {N}^+}\) and \((c_n)_{n\in \mathbb {N}^+}\) such that \(a_n=\lfloor \ln (n)\rfloor \) and \(c_n=\lfloor c\ln (n)\rfloor \). Let

$$\begin{aligned} I\,:=\inf \{t\in \mathbb {N}^+: S_i(t) = c_n \text { for some } i\in [n] \}, \end{aligned}$$

the first time that there exists a coupon type for which exactly \(c_n\) copies have been collected. Then there exists \(\delta >0\) such that

$$\begin{aligned} \mathbb {P}(|R_{I}|\le a_n)=O(n^{-\delta }), \qquad n \rightarrow \infty . \end{aligned}$$

Proof

We assume, in all that follows, that n is sufficiently large that \(c_n>1\). For \(k\in [n]\), let

$$\begin{aligned} I_k=\inf \{t\in \mathbb {N}^+: S_{k}(t) =c_n\} \end{aligned}$$

be the first time that \(c_n\) copies of coupon type \(D_k\) have been collected. For any sequence of natural numbers \((b_n)_{n\in \mathbb {N}^+}\), we have

$$\begin{aligned} \mathbb {P}(|R_{I_k}| \le a_n)&= \mathbb {P}(|R_{I_k}|\le a_n, I_k\le b_n) + \mathbb {P}(|R_{I_k}|\le a_n, I_k> b_n)\nonumber \\&\le \mathbb {P}(I_k\le b_n) + \mathbb {P}(|R_{I_k}|\le a_n, I_k> b_n)\nonumber \\&\le \mathbb {P}(I_k\le b_n) + \mathbb {P}(W_{a_n+1}>b_n), \end{aligned}$$
(B.5)

where the last inequality follows by observing that if \(|R_{I_k}|\le a_n\) and \(I_k>b_n\), then \(W_{a_n+1}>b_n\).

To find an upper bound for \(\mathbb {P}(I_k\le b_n)\), note that, for any \(s\ge 1\), the random time between the sth and \((s+1)\)th arrival of coupon type \(D_k\) is a geometric random variable with success probability 1 / n. It follows that \(\varDelta _k\,:=I_k-W_k\) is a sum of \(c_n-1\) independent geometric random variables,Footnote 6 each with success probability 1 / n. Lemma B.4 therefore implies that for any \(0<\lambda <1\),

$$\begin{aligned} \mathbb {P}(\varDelta _k\le \lambda n(c_n-1)) \le e^{-f(\lambda ) c_n + f(\lambda )} \end{aligned}$$

where \(f(\lambda )>0\). But from the trivial lower bound \(W_k\ge 1\), it follows that \(\varDelta _k\le I_k -1\). Therefore, for any \(b_n\le \lambda n(c_n-1)+1\), we have

$$\begin{aligned} \mathbb {P}(I_k \le b_n) \le \mathbb {P}(I_k\le \lambda n (c_n-1) + 1) \le \mathbb {P}(\varDelta _k\le \lambda n (c_n-1)) \le e^{-f(\lambda ) c_n + f(\lambda )}. \end{aligned}$$
(B.6)

To find an upper bound for \(\mathbb {P}(W_{a_n+1}>b_n)\), we begin with the observation that, with the convention \(W_0=0\), we have

$$\begin{aligned} W_{a_n+1} = \sum _{i=0}^{a_n}(W_{i+1}-W_i). \end{aligned}$$

For \(0\le i \le a_n\), the random variables \(W_{i+1}-W_i\) are independent, and distributed according to a geometric distribution with success probability \(1-i/n\). Therefore, Lemma B.4 implies that for any \(\zeta >1\)

$$\begin{aligned} \mathbb {P}(W_{a_n+1}\ge \zeta \,\mathbb {E}[W_{a_n+1}]) \le e^{-f(\zeta )(1-a_n/n)\mathbb {E}(W_{a_n+1})} \end{aligned}$$

with \(f(\zeta )>0\). But explicit calculation shows that

$$\begin{aligned} \mathbb {E}(W_{a_n+1}) = n(H_n - H_{n-a_n-1})\sim a_n, \qquad n\rightarrow \infty , \end{aligned}$$

where \(H_i\) is the ith harmonic number, and the asymptotic result follows from \(H_n\sim \ln (n)\) and the fact that \(a_n=o(n)\). It follows that for any choice of \(b_n \ge \zeta \, \mathbb {E}(W_{a_n+1})\) and \(\alpha \in (0,f(\zeta ))\), for sufficiently large n, we have

$$\begin{aligned} \mathbb {P}(W_{a_n+1}\ge b_n) \le e^{-\alpha \, a_n}. \end{aligned}$$
(B.7)

Any choice of \(b_n\) satisfying \(\zeta \,\mathbb {E}(W_{a_n+1}) \le b_n \le \lambda n(c_n-1)+1\), for sufficiently large n, suffices to ensure (B.6) and (B.7) hold simultaneously. It therefore suffices to set \(b_n=n\). For simplicity, \(\lambda \in (0,1)\) and \(\zeta >1\) can be chosen so that \(f(\lambda )=1=f(\zeta )\). Combining (B.5), (B.6) and (B.7) then implies that for any \(\alpha <1\) we have

$$\begin{aligned} \mathbb {P}(|R_{I_k}| \le a_n) \le e^{-c_n+1} + e^{-\alpha \,a_n} \end{aligned}$$

for sufficiently large n.

Finally, since \(|R_I|\le a_n\) implies \(|R_{I_k}|\le a_n\) for some \(1\le k \le a_n\), it follows from the union bound that, for sufficiently large n,

$$\begin{aligned} \mathbb {P}(|R_I|\le a_n) \le \mathbb {P}\left( \bigcup _{k=1}^{a_n}\{|R_{I_k}|\le a_n\}\right) \le \sum _{k=1}^{a_n} \mathbb {P}(|R_{I_k}|\le a_n)&\le a_n\, e^{-c_n+1} + a_n\, e^{-\alpha \,a_n}\\&\le e^2\,\ln (n)\,n^{-c} + e^{\alpha }\,\ln (n)\,n^{-\alpha }. \end{aligned}$$

Since \(c,\alpha >0\), we can choose \(0< \delta <\min \{c,\alpha \}\), and we obtain \(\mathbb {P}(|R_I|\le a_n) = O(n^{-\delta })\).   \(\square \)

Lemma B.4

Let \(X_1,X_2,\ldots ,X_n\) be independent random variables, such that \(X_i\) has geometric distribution with success probability \(p_i\), and let \(X=\sum _{i=1}^n X_i\). Then

$$\begin{aligned} \mathbb {P}(X\le \lambda \mu )&\le e^{-p_{*}\mu f(\lambda )}, \qquad \forall \,\lambda \le 1,\\ \mathbb {P}(X\ge \zeta \mu )&\le e^{-p_{*}\mu f(\zeta )}, \qquad \forall \,\zeta \ge 1, \end{aligned}$$

where \(\mu =\mathbb {E}(X) = \sum _{i=1}^n 1/p_i\), \(p_*= \min _{i\in [n]} p_i\) and \(f(x)=x -1 -\ln (x)\).

Proof

These results can be established, in the standard way, by applying Markov’s inequality to \(\mathbb {E}(e^{t X})\), and using the explicit form for \(\mathbb {E}(e^{t X_i})\); see e.g. [31]. \(\square \)

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Collevecchio, A., Elçi, E.M., Garoni, T.M. et al. On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model. J Stat Phys 170, 22–61 (2018). https://doi.org/10.1007/s10955-017-1912-x

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Keywords

  • Coupling from the past
  • Relaxation time
  • Random–cluster model
  • Markov-chain Monte Carlo