Entropy-Driven Cutoff Phenomena

Abstract

In this paper we present, in the context of Diaconis’ paradigm, a general method to detect the cutoff phenomenon. We use this method to prove cutoff in a variety of models, some already known and others not yet appeared in literature, including a non-reversible random walk on a cylindrical lattice. All the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon. In the case of birth-and-death chains this mechanism is fairly well understood; our work is an effort to generalize this picture to more general systems, such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one-dimensional problem. In those situations the drift may be looked for by means of a suitable partitioning of the state space into classes; using a statistical mechanics language it is then possible to set up a kind of energy-entropy competition between the weight and the size of the classes. Under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease.

Appendices

Appendix A: Mean Value and Variance of ${\zeta_{n}^{1}}$ for the Mean-Field Ising Model

In this appendix we present in full details the estimates for ${\mathbb{E}[{\zeta_{n}^{1}}]}$ and $\text{Var}[{\zeta_{n}^{1}}]$ we have used to apply Corollary 1.3 to the magnetization chain in Sect. 3.6. Since for β=0 the magnetization chain reduces to the Ehrenfest chain, the following estimates hold as well for the Ehrenfest Urn model presented in Sect. 3.3.

Standard formulas (see e.g. [2]) give

(A.1)

(A.2)

where ζ _k→k−1 is the first time the chain visits k−1 after visiting k and

(A.3)

(A.4)

Let us begin rewriting the ratio of the two binomial coefficients as

(A.5)

Next, note that for any of the values of triple (i,j,k) involved in the calculations

$$ 0 \leq\frac{i}{\frac{n}{2}+k+1} \leq\frac{i}{\frac{n}{2}-k} \leq1 $$

(A.6)

So we find handy the following two easy lemmas.

Lemma A.1

For x∈[0,1]

$$ (1-x)\frac{1}{1+x} \leq e^{-2x} $$

(A.7)

Lemma A.2

For 0≤y≤x≤1

$$ (1-x)\frac{1}{1+y} \leq e^{-x-y} $$

(A.8)

In virtue of Lemma A.2 we can bound line (A.5) as follows:

(A.9)

(A.10)

(A.11)

Thus, for ${\frac{1}{2}\sqrt{\frac{n}{1-\beta}}}\leq k \leq\frac {n}{2}-\log n$,

(A.12)

(A.13)

(A.14)

(A.15)

(A.16)

Therefore we obtain the following upper bounds:

(A.17)

(A.18)

and

(A.19)

(A.20)

From previous computations, noticing that

$$ e^{\frac{-2(1-\beta)l^2 + 2l }{ n ( 1-\frac{4k^2}{n^2} + \frac{2}{n} - \frac{4k}{n^2} ) }} \leq\sqrt{e} $$

(A.21)

we have that

$$ {\mathbb{E}[\zeta_{k\to k-1}]} \leq\sqrt{e} \frac{{1+e^{\frac {4\beta k}{n}}}}{2(1-\beta )}\frac{n}{k} $$

(A.22)

and then by summation

$$ {\mathbb{E}[\zeta_{k+l\to k-1}]} \leq\frac{\sqrt{e}}{2(1-\beta} n \log \biggl(1+\frac{l}{k} \biggr) + O(n) $$

(A.23)

From (A.2), using (A.11) and (A.22)–(A.23), we can easily bound the variance of ζ _k→k−1 as follows:

(A.24)

(A.25)

(A.26)

(A.27)

(A.28)

(A.29)

Therefore $\text{Var}[{\zeta_{n}^{1}}] = \sum_{k={\frac{1}{2}\sqrt {\frac{n}{1-\beta}}}+1}^{\frac{n}{2}} \text{Var}[\zeta_{k\to k-1}]$ grows at most as O(n ²).

Eventually, let us bound from below the expectation ${\mathbb{E}[{\zeta_{n}^{1}}]}$. From (A.1) and (A.5) we have

(A.30)

Then we have

(A.31)

(A.32)

where ε ₁ tends to 0 exponentially fast in n.

Remark A.1

The error ε ₁ gives a negligible contribution to ${\mathbb{E}[{\zeta_{n}^{1}}]}$ being exponentially small, for this reason we will henceforth drop it.

The right-hand in (A.32) can be rewritten as follows

(A.33)

(A.34)

(A.35)

with ε ₂=o(n ⁻¹). Now set $\varphi= -\frac{4k^{2}}{n^{2}}+\frac{2}{n}-\frac{4k}{n^{2}}$, then (A.35) can be rewritten as follows

(A.36)

(A.37)

(A.38)

where ε=O(log⁻²(logn)) and $\varepsilon_{3} = O ( n^{-\frac{1}{2}}\log\log n )$.

Therefore

(A.39)

where

$$ \gamma= \biggl[\frac{1-\frac{\log n}{\sqrt{n(1-\beta)}}}{1+\frac{\log n}{\sqrt{n(1-\beta)}}} \biggl( 1+2\beta \frac{\log n}{\sqrt{n(1-\beta)}} + O \biggl( \frac{\log^2n}{n} \biggr) \biggr) \biggr]^{1+\frac{\sqrt{n}}{\log\log n}} $$

(A.40)

and

$$ \varGamma= \frac{{\frac{n}{2}+k}}{\frac{n}{2}(\frac{-4\beta k}{n}+O ( \log^{-2}n )) + k(2+\frac{4\beta k}{n} + O ( \log^{-2}n ))+2} $$

(A.41)

Now, γ can be rewritten as

$$ \gamma= \biggl[ 1-\frac{2(1-\beta)\log n}{\sqrt{n(1-\beta)}} + O \biggl( \frac{\log^2n}{n} \biggr) \biggr]^{1+\frac{\sqrt{n}}{\log\log n}} $$

(A.42)

Therefore γ tends asymptotically to 0.

The right-hand in (A.39) now becomes

$$ (1-\gamma) (1-\varepsilon)\sum_{{\frac{1}{2}\log n\sqrt{\frac {n}{1-\beta}}}}^{{\frac{n}{\log n}}} \frac{2n ( 1+O ( \log^{-1}n ) ) }{2k(1-\beta ) + 2 + O ( \log^{-1}n )} $$

(A.43)

from which we see that, to the leading order in n

$$ {\mathbb{E}\bigl[{\zeta_n^{1}}\bigr]} \geq \frac{1}{2(1-\beta)}n\log n $$

(A.44)

Appendix B: Mean Value and Variance of ${\zeta_{n}^{1}}$ for the Partially Diffusive Random Walk

Standard formulas (see e.g. [2]) give

(B.1)

where ζ _k→k−1 is the first time the chain visits k−1 after visiting k. By means of (3.46)–(3.48) and reversibility,

(B.2)

(B.3)

(B.4)

Using the properties of the exponential integral we get

$$ \phi(k) = \frac{1}{\log2} - \frac{k}{n\log2}2^{(k-n)} + O \biggl(\frac{1}{k} \biggr) $$

(B.5)

and therefore

$$ {\mathbb{E}\bigl[{\zeta_n^{1}}\bigr]} = \frac{2(1-\varepsilon)}{\log2} n \log n + O \bigl( n^{1-\varepsilon} \bigr) $$

(B.6)

Similarly, for n sufficiently large we have that

$$ {\mathbb{E}\bigl[{\zeta_n^{1}}-{ \zeta_n^{\theta}}\bigr]} = \sum_{k=n^\varepsilon+ 1}^{n^\varepsilon\theta^{n^{2\varepsilon-1}}} \frac{2n}{k} \phi(k) = \frac{2 n^{2\varepsilon}}{\log2} \log\theta+ O \bigl( n^\varepsilon\log\theta \bigr) $$

(B.7)

From (B.7) we see that for n sufficiently large ${\mathbb{E}[{\zeta_{n}^{1}}-{\zeta_{n}^{\theta}}]}$ grows as n ^2ε at most.

To compute $\text{Var}[{\zeta_{n}^{1}}]$ we use the following formulas

(B.8)

(B.9)

Then we estimate the sum from below as its first term

(B.10)

and from above as

(B.11)

(B.12)

(B.13)

(B.14)

(B.15)

From (B.10) and (B.15) we see that $\text{Var}[\zeta_{k\to k-1}] = O ( \frac{n^{2}}{k^{2}} )$ and therefore, to the leading order, $\text{Var}[{\zeta_{n}^{1}}] = O ( n^{2-\varepsilon} )$.