Monotonicity properties of exclusion sensitivity

In~\cite{bgs2013}, exclusion sensitivity and exclusion stability for symmetric exclusion processes on graphs were defined as a natural analogue of noise sensitivity and noise stability in this setting. As these concepts were defined for any sequence of connected graphs, it is natural to study the monotonicity properties of these definitions with respect to adding edges to the graphs, and in particular, whether some graphs are more stable or sensitive than others. The main purpose of this paper is to answer some such question from~\cite{bgs2013}. The main tool used is included results about the eigenvectors and eigenvalues of the generator of symmetric exclusion processes on complete graphs.


Introduction
The notions noise sensitivity and noise stability was first introduced in [1], decribing how sensitive a sequence of Boolean functions f n : {0, 1} n → {0, 1} was to a particular kind of noise in the argument. The main type of noise considered was resampling each entry in its argument of f n with a small probability, or equivalently, be letting each such entry run a continuous time Bernoulli process in a short time interval. Since this paper was published, similar definitions have been made in slight diifferent settings, by changing one or several of the elements in the setup, such as the domain of the functions f n , the range of the functions f n or the process constituating the noise ( [1], [4], [3]). One such generalization was given in [2], where the range and domain of the functions (f n ) n≥1 was kept from the original setting, but the process was changed into a symmetric exclusion process with respect to some sequence of connected graphs, (G n ) n≥1 . In this new setting, it is natural to ask to what extent the sensitivity of a sequence of functions depends on the sequence of graphs. This is the main subject of this paper.
We now define what we mean by a symmetric exclusion process. Let (G n ) n≥1 be an increasing sequence of finite connected graphs and let (α n ) n≥1 be a sequence of real numbers such that α n . In this paper, we are interested in the sequence of Markov chains (X (n) ), where X (n) is a symmetric exclusion process on {0, 1} V (Gn) with rate α n . This process can be defined as follows. At time zero, put a black or white marble at each vertex of the graph. Now for each edge e ∈ E(G n ), associate an independent Poisson clock with rate α n . When this clock rings, we interchange the marbles at the endpoints of e. Let X (n) t be the configuration of marbles at time t.
In general, for any graph G n , we will identify confgurations of black and white marbles with elements in {0, 1} V (Gn) by letting the numbers be indicators of black marbles. Similarly, for each ℓ ∈ {0, 1, . . . , |V (G n )|}, we will identify elements in V (Gn) ℓ with configurations with exactly ℓ black marbles by representing such a configuration by the set of vertices on which there is black marbles.
It is easy to check that the uniform distribution π n on {0, 1} V (Gn) will be a stationary distribution for the symmetric exclusion process X (n) on G n with rate α n . However, as the Markov process X (n) is not irreducible, this is not the only stationary distribution, and in fact for any ℓ ∈ 0, . . . , |V (G n )|, the uniform distribution π (ℓ) n on V (Gn) ℓ will be a stationary distribution for X (n) . For x, y ∈ V (G) ℓ , write x ∼ y to denote that y can be obtained from x by interchanging the marbles at the endpoints of some edge in E(G). Whenever we pick X (n) 0 ∼ π n , we will say that X (n) is a symmetric exclusion process with respect to (G n , α n ) n≥1 , and whenever we pick X (n) 0 ∼ π (ℓn) n we will say that X (n) is a symmetric exclusion process with respect to (G n , α n , ℓ n ) n≥1 .
We now give the two definition from [2] with which we will be concerned.
Definition 1.1. Let (G n ) n≥1 be an increasing sequence of finite connected graphs and let (α n ) n≥1 be a sequence of real numbers. For each n ≥ 1, let X (n) be the symmetric exclusion process on {0, 1} V (Gn) with rate α n where X (n) 0 ∼ π n . The sequence of functions The next definition captures the opposite behaviour.
be an increasing sequence of finite connected graphs and let (α n ) n≥1 be a sequence of real numbers. For each n ≥ 1, let X (n) be the symmetric exclusion process on {0, 1} V (Gn) with rate α n where X (n) 0 ∼ π n . The sequence of functions In addition to the two definitions above, a sequence of functions f n : {0, 1} V (Gn) → {0, 1} is said to be complete graph exclusion sensitive (CGXS) is it is exclusion sensitive with respect to (K n , 1/n) n≥1 and complete graph exclusion stable (CGXStable) if it is exclusion stable with respect to (K n , 1/n) n≥1 .
It is relatively easy to see that any sequence of functions (f n ) n≥1 for which lim n→∞ Var(f n (X (n) 0 )) = 0 will be both exclusion stable and exclusion sensiitiive with respect to (G n , α n ) n≥1 for any sequence (α n ) n≥1 of positive numbers. For this reason, we will only be interested in so called nondegenerate sequences of functions (f n ) n≥1 , meaning that Var(f n (X (n) 0 )) is uniformly bounded away from zero.
In [2], the authors asked the following two questions. For a sequence (G n , α n ) n≥1 , with α n ≤ 1/ max v∈V (Gn) deg v, and a sequence of functions f n : Neither of these questions were confirmed or proven to be false in their paper. The main objective of this paper is to provide a proof of the following result, which provides a positive answer to both questions.
1} be a sequence of real valued functions, and let (G n ) n≥1 be a sequence of connected graphs. Further let α n ≤ 1/ max v∈V (Gn) deg v. Then Given the positive answers to both questions above, one might ask if being XS (or XStable) is monotone with respect to adding edges to the graphs (G n ) n≥1 . We will later see that this is true if we use the same rates for both graphs, but the following example shows that the restriction α n ≤ 1/ max v∈V (Gn) deg v is not strong enough to get monotonicity. Example 1.4. Let G n be the graph with vertex set {1, 2, . . . , 2n} and an edge between two vertices i and j if and only if |i − j| = 1 mod n. Further, let G ′ n be the graph obtained from G n by adding the edge (i, j) for all i, j ∈ {n + 1, n + 2, . . . , 2n}. Finally, let G ′′ n = K 2n . For x ∈ {0, 1} V (Gn) , define f n (x) = (−1) |x∩{1,3,...n}| . Then (f n ) n≥1 is exclusion sensitive with respect to (G n , 1/2) n≥1 , exclusion stable with respect to (G ′ n , 1/(n − 1)) n≥1 and exclusion sensitive with respect to (G ′′ n , 1/(n − 1)) n≥1 .   We now give a somewhat more complicated example, which shows that assuming regularity is not enough to obtain monotonicity, in the sense that there is two sequences of regular graphs, (G n ) n≥1 and (G ′ n ) n≥1 with the same vertex sets and with E(G n ) ⊂ E(G ′ n ) for all n, and a sequence of Boolean functions f n : and both G n and G ′ n are vertex transitive for each n, but (f n ) n≥1 is exclusion sensitive with respect to (G n , 1/2) n≥1 and exclusion stable with respect to (G ′ n , 1/n) n≥1 , where the rates in both cases are chosen to be the inverse of the maximal degree in the given graph.
The rest of this paper will be structured as follows. In the next section, we give notation for the eigenvectors and eigenvalues for the different processes with which we will be concerned in this paper. We also give spectral equivalences of exclusion stability and exclusion sensitivity.
In the third section we study the structure of these eigenvectors and eigenvalues a bit closer. In particular, we prove some results concerning these for the symmetric exclusion process with respect to (K n , α n ) n≥1 for some sequence (α n ) n≥1 of real numbers. In the fourth section, we give a proof of our main result. Finally, in the last section, we give a monotonicity result.

Spectral equivalences of exclusion sensitivity and stability
All of the results presented in this paper will use Fourier analysis. In this section we will define the functions which we will use as a basis, and derive some simple results.
Let X (n,ℓ) be a symmetric exclusion process with respect to (G n , α, ℓ). Then X (n,ℓ) is a Markov process and has a generator Q )] is an inner product. As X (n,ℓ) is reversible and irreducible, we can find a set {ψ (1) such that {ψ (n,ℓ) i } i is an orthonormal basis with respect to ·, · for the space of real valued functions on V (Gn) ℓ . Note that we can assume that ψ (n,ℓ) 1 ≡ 1.
Next, for all t ≥ 0, let H In other words, H The eigenvectors {ψ (n,ℓ) i } i will be eigenvectors to H (n,ℓ) t as well, with corresponding eigenvalues To simplify notations, we will writef (ℓ) (i) instead of f, ψ (n,ℓ) i . Using these Fourier coefficients, for any function f : and Var(f (X (n,ℓ) 0 Another characterization of the eigenvalues which will be useful for us later is where the minimum is attained by the corresponding eigenvector ψ (n,ℓ) i . The ratio on the right hand side of (2) is called the Rayleigh quotient of −Q (ℓ) n . It is easy to see that if ψ is an eigenvector of −Q n , the Rayleigh quotient is the corresponding eigenvalue. From this, the linearity of −Q (ℓ) n and the bilinearity of ·, · , it follows that it follows that if Q n is the generator of the exclusion process with respect to (G n , α, ℓ) and Q ′ n the the generator of the exclusion process with respect to (G ′ n , α, ℓ) for some graph G ′ n satisfying This fact will also be useful later.
Above, we listed some simple properties the eigenvectors of the generator of an exclusion process with a fixed number of black marbles. The next lemma relates these the the eigenvectors of an exclusion process with a random number of black marbles. Below and in the rest of these notes, whenever f, g : be an increasing sequence of finite connected graphs and let (α n ) n≥1 be a sequence of positive real numbers. For each ℓ ∈ {0, 1, . . . , |V (G n )|}, let Q (ℓ) n be the generator of the exclusion process with respect to (G n , α n , ℓ), and let {ψ i,ℓ } i,ℓ is an orthonormal basis for −Q n , where Q n is the generator of the symmetric exclusion process with respect to (G n , α n ), with corresponding eigenvalues {λ This shows that ψ . The claims of orthonormality follows similarly, and is therefore omitted here. Remark 2.3. Note that the eigenvectors given by (3) with corrsponding eigenvalue equal to one is independent on the chosen graph G n as long as G n is connected.
Whenever we calculate the Fourier coefficients of some function f : The next lemma provides a spectral characterization of what it means to be exclusion sensitive, and it is the equivalent definition it provides that we will use in all subsequent results.
be an increasing sequence of finite connected graphs, let (α n ) n≥1 be a sequence of positive real numbers and for each n ≥ 1, let X (n) be the exclusion process with respect to (G n , α n ) n≥1 . Further let {ψ i,ℓ } i,ℓ be the corresponding eigenvalues. Then a sequence Proof. First recall the well known result stating that for three random variables X, Y and Z, We will now rewrite the term Cov f (X = ℓ) in the expression above. To this end, note that for any ℓ ∈ {0, 1, . . . , n}, Here the last equality follows from the fact that {ψ Note in particular that the term e −ελ (n) i,ℓf n (i, ℓ) 2 . in the previous equation is positive. From this fact and (4), it follows that (f n ) n≥1 can be XS with respect to (G n , α n ) n≥1 if and only if For any ε > 0, it is easy to see that (ii) is satisfied if and only if (ii') holds. From this the desired conclusion follows.
The following lemma provides an analogue of Lemma 2.4 for exclusion stability.
Lemma 2.5. Let (G n ) n≥1 be an increasing sequence of finite connected graphs, let (α n ) n≥1 be a sequence of positive real numbers and for each n ≥ 1, let X (n) be the exclusion process with respect to (G n , α n ) n≥1 . Further let {ψ Proof. First note that since f n is Boolean, we have that Using this, as well as the the proof of the previous proposition, we obtain For the if direction of the proof, suppose that there for any δ > 0 is k ≥ 1 such that Then for all δ > 0, As δ can be chosen to be arbitrarily small, this implies that (f n ) n≥1 is exclusion stable with respect to (G n , α n ) n≥1 . For the only if direction, suppose that there is δ > 0 such that for all k ≥ 1, for all k > 0. Then in particular, this is true for k = ε −1 . This implies that In particular, (f n ) n≥1 cannot be noise stable.
Before ending this section, we present a last lemma which gives an upper bound of the eigenvalues λ Lemma 2.6. Let Q (ℓ) be the generator of the symmetric exclusion process on G with ℓ black marbles and rate α. If λ is an eigenvalue of −Q (ℓ) and d := max v∈V (G) deg v, then λ ≤ 2αℓd.
Proof. By using the Rayleigh quotient, we obtain

Eigenvectors and eigenvalues for symmetric exclusion processes
Below and in the rest of this section, for any graph G, any ℓ ∈ {0, 1, . . . , |V (G)|}, any x ∈ V (G) ℓ and any v ∈ V (G), let x v denote the unique element in V (G) ℓ−1 or V (G) ℓ+1 which differ from x in only the color at vertex v. Moreover, for any e ∈ E(G), let x e be the unique element in which is obtained by switching positions of the marbles at the endpoints of e. For any v ∈ V (G), let Finally, for m < ℓ and y ∈ V (G) Our main objective in this section will be to give a proof of the following result, which will play a major role in the later proof of out main result, Proposition 1.3.
Then (a) for each j = 1, . . . , ℓ, the eigenvalue αj(n − j + 1) has multiplicity n j − n j−1 , (b) If we order the eigenvalues of −Q Before giving a proof of Proposition 3.1, we will state and prove three rather technical lemmas which we will need for the proof. The first of these lemmas provide a way to obtain eigenvectors of −Q (ℓ−1) and −Q (ℓ+1) given an eigenvectors of −Q (ℓ) . Lemma 3.2. Let G be a finite connected graph, ℓ ∈ N and α any positive real number. Let Q (ℓ) be the generator of the symmetric exclusion process with respect to (G, α, ℓ), and let ψ be an eigenvector to −Q (ℓ) with eigenvector λ. Then, for x ∈ V (Gn) ℓ−1 , the function ψ + : V (Gn) Proof. Note first that for any x ∈ {0, 1} V (Gn) , Using this, for x ∈ V (Gn) ℓ−1 , we obtain This shows that ψ + is an eigenvector to −Q (ℓ−1) with eigenvalue λ, provided that ψ + ≡ 0.
in turn implying that As this shows that ψ − is an eigenvector to −Q (ℓ+1) with eigenvalue λ provided that ψ − ≡ 0, this concludes the proof.
The purpose of the next lemma is to provide expressions for the lengths of ψ + and ψ − . In contrast to the previous lemma this lemma requires that the graph G n on which the exclusion process evolves is the complete graph. and Proof. In what follows, for x, x ′ ∈ V (Gn) ℓ , let x ∼ x ′ denote that it is possible to get from x to x ′ by exchanging the position of two marbles. With this notation, we can derive an expression for ψ + , ψ + as follows. Analogously, The next lemma shows that orthogonality is preserved by the operations ψ → ψ + and ψ → ψ − . n be the generator of the symmetric exclusion process with respect to (K n , α). Then for any two orthogonal eigenvectors ψ and ψ ′ to −Q (ℓ) Proof. By reasoning as in the proof of Lemma 3.3, we obtain Analogously, By the orthogonality of ψ and ψ ′ , the desired conclusion follows.
We are now ready to give a proof of Proposition 3.1.
Proof of Proposition 3.1. We will first prove that (a), (b) and (c) holds by using induction och the number of black marbles, ℓ. As induction hypothesis, suppose that for some ℓ ∈ N, there is an orthonormal basis of eigenvectors ψ . By Lemma 3.3 and the induction hypothesis, for any i ∈ {1, 2, . . . , n ℓ }, To show that the induction hypothesis must hold for ℓ + 1 black marbles given that it holds for ℓ black marbles, it now suffices to show that λ By the induction hypothesis, this is nonzero for i ∈ {1, 2, . . . , n ℓ }. As no orthogonal set of eigenvectors to −Q (ℓ) n can contain more than n ℓ elements, we must have that λ (n,ℓ+1) i = α(ℓ + 1)(n − ℓ) for all i > n ℓ . As the induction hypothesis is well known to hold for ℓ = 0, the desired conclusion follows.

A proof of the main result
Before we give a proof of our main result, Proposition 1.3, we will prove the following lemma, which is interesting in it self, relating the eigenvectors of an exclusion process on any graph with the eigenvectors of an exclusion process on the complete graph. Then for any k < n/2 and k ′ such that αk ′ (n − k ′ + 1) ≥ j for all j with αj(n − j + 1) ≤ k, Consequently, if Q n is the generator of the symmetric exclusion process on K n with rate α n , R n is the generator of the symmetric exclusion process of G n with rate β n and {ψ i,ℓ } are the orthonormal bases of eigenvectors of −Q n and −R n respectively, as defined by Lemma 2.2, then When we use Lemma 4.1 in the proof of Proposition 1.3, we will think of n as being very large and k as being small and fixed, and pick α = 1/n and β = 1/d. With this choice of parameters, and any k, k ′ ≤ n/2 such that k ′ (n − k ′ + 1) ≥ jn for all j with j(n − j + 1) ≤ kn, Lemma 4.1 says that valid for x ∈ [0, n/2], we obtain that in this special case, we can choose any k ′ ≥ 2n n+1 2 · k. In particular, we can choose k ′ = 4k. From this we get the following lemma as a corollary.
n be the generator of the symmetric exclusion process on K n with ℓ black marbles and rate 1/n, and let R for all k ≤ n/8. for any k ′ such that αk ′ (n − k ′ + 1) ≥ j for all j with αj(n − j + 1) ≤ k.
Proof of Proposition 1.3. Note first that it is enough to prove the result for β n = 1/ max v∈V (Gn) deg v.
Suppose that (f n ) n≥1 is not exclusion sensitive with respect to the sequence (K |V (Gn)| , 1/|V (G n )|) n≥1 . By Lemma 2.4, either lim or there is k > 0, ε > 0 and a subsequence n ′ such that i,ℓ : 0<λ for all n ′ . In the first case, we are already done, so we can assume that second of these is the case. By Lemma 4.2, this implies that i,ℓ : 0<µ i,ℓ . This implies that (f n ) n≥1 is not exclusion sensitive with respect to (G n , 1/d n ) n≥1 , and finishes the proof of (i).
To show that (ii) holds, suppose that (f n ) n≥1 is exclusion stable with respect to (K |V (Gn)| , 1/|V (G n )|) n≥1 . Then, by Lemma 2.5, for all δ > 0 there is k > 0 such that or equivalently, such that inf By Lemma 4.2, this implies that As δ was arbitrary, by Lemma 2.5, (f n ) n≥1 is exclusion stable with respect to (G n , 1/d n ) n≥1 . This finishes the proof.

Monotonicity at equal rate
The main purpose of this section is to give a proof of the following result, which shows that for any fixed sequence of rates (α n ) n≥1 , any sequence of graphs (G n ) n≥1 and any sequence of functions f n : V (G n ) → {0, 1}, the properties of being XS and XStable with respect to (G n , α n ) n≥1 are monotone to adding edges to the graphs in (G n ) n≥1 .
Proposition 5.1. Let (G n ) n≥1 and (G ′ n ) n≥1 be two sequences of finite connected graphs with V (G n ) = V (G ′ n ) and E(G ′ n ) ⊆ E(G n ), and let α n be a sequence of real numbers. Let f n : {0, 1} V (Gn) → {0, 1} be a nondegenerate sequence of functions. Then To be able to formulate and prove the lemma we will use in the proof of Proposition 5.1, we will need some more notation. Given a graph G and a positive real number α, let Q be the generator of the symmetric exclusion process with respect to (G, α). Let {ψ i,ℓ } i,ℓ be the set of eigenvectors and {λ i,ℓ } i,ℓ be the corresponding set of eigenvalues of −Q. For each real valued function f with domain {0, 1} V (G) we define the spectral probability measure P f Lemma 5.2. Let Q be the generator for the symmetric exclusion process with respect to (G, α) and Q ′ be the generator for the symmetric exclusion process with respect to (G ′ , α), for two finite connected graphs G and G ′ with the same number of vertices and some positive real number α. Let {ψ i,ℓ } i,ℓ be and orthonormal set of eigenvectors to −Q with corresponding eigenvalues {λ i,ℓ } i,ℓ , and let {χ i,ℓ } i,ℓ and {µ i,ℓ } i,ℓ be the corresponding sets for −Q ′ . Further, let f : {0, 1} V (G) → R and let P f and P ′ f be the spectral probability measures with respect to Q and Q ′ respectively. Then, if E(G ′ ) ⊆ E(G), for all positive real numbers we have that Proof. For some fixed value k whose exact value will be decided upon later, let P λ≤k f := i,ℓ : 0<λ i,ℓ ≤k f, ψ i,ℓ ψ i,ℓ be the projection of f onto the space spanned by all eigenvectors ψ i,ℓ with eigenvalue less than or equal to k but not equal to zero. Set P λ>k f := i,ℓ : λ i,ℓ >k f, ψ i,ℓ ψ i,ℓ . Then for any k ′ > 0, where the last equality follows from Remark 2.3. Using first that any eigenvector with corresponding eigenvalue equal to zero is orthogonal to any eigenvector χ i,ℓ with corresponding eigenvalue µ i,ℓ ≥ k ′ > 0, and then Cauchy-Schwarz inequality, we obtain P f (µ > k ′ ) = i,ℓ : µ i,ℓ >k ′ P λ≤k f + P λ>k f, χ i,ℓ 2 Var f = i,ℓ : µ i,ℓ >k ′ P λ≤k f, χ i,ℓ 2 + 2 P λ≤k f, χ i,ℓ P λ>k f, χ i,ℓ + P λ>k f, Here i,ℓ : µ Var P λ≤k f Var f .
Using that where the second inequality follows by Remark 2.1. For the second term in (6), we obtain i : µ (n) i,ℓ >k ′ P λ>k f, χ (n) i,ℓ 2 Var f = Var P λ>k f Var f = P f (λ > k).
Summing up, we thus get, which is the desired conclusion.
We now give a proof of Proposition 5.1.
Lemma 2.4 now ensures that (f n ) n≥1 is exclusion sensitive with respect to (G n , α n ) n≥1 .