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In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), due to Template:Harvs. Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated.

An earlier version, for the special case of i.i.d. variables, called Harris inequality, is due to Template:Harvs, see below. One generalization of the FKG inequality is the Holley inequality (1974) below, and an even further generalization is the Ahlswede–Daykin "four functions" theorem (1978). Furthermore, it has the same conclusion as the Griffiths inequalities, but the hypotheses are different.

The inequality

Let $X$ be a finite distributive lattice, and μ a nonnegative function on it, that is assumed to satisfy the (FKG) lattice condition (sometimes a function satisfying this condition is called log supermodular) i.e.,

$\mu (x\wedge y)\mu (x\vee y)\ge \mu (x)\mu (y)$

for all x, y in the lattice $X$.

The FKG inequality then says that for any two monotonically increasing functions ƒ and g on $X$, the following positive correlation inequality holds:

$(\sum _{x\in X}f(x)g(x)\mu (x))(\sum _{x\in X}\mu (x))\ge (\sum _{x\in X}f(x)\mu (x))(\sum _{x\in X}g(x)\mu (x)).$

The same inequality (positive correlation) is true when both ƒ and g are decreasing. If one is increasing and the other is decreasing, then they are negatively correlated and the above inequality is reversed.

Similar statements hold more generally, when $X$ is not necessarily finite, not even countable. In that case, μ has to be a finite measure, and the lattice condition has to be defined using cylinder events; see, e.g., Section 2.2 of Template:Harvtxt.

For proofs, see the original Template:Harvtxt or the Ahlswede–Daykin inequality (1978). Also, a rough sketch is given below, due to Template:Harvtxt, using a Markov chain coupling argument.

Variations on terminology

The lattice condition for μ is also called multivariate total positivity, and sometimes the strong FKG condition; the term (multiplicative) FKG condition is also used in older literature.

The property of μ that increasing functions are positively correlated is also called having positive associations, or the weak FKG condition.

Thus, the FKG theorem can be rephrased as "the strong FKG condition implies the weak FKG condition".

A special case: the Harris inequality

If the lattice $X$ is totally ordered, then the lattice condition is satisfied trivially for any measure μ. For this case, the FKG inequality is Chebyshev's sum inequality: if the two increasing functions take on values $a_1\le a_2\le \cdots \le a_n$ and $b_1\le b_2\le \cdots \le b_n$, then (we may assume that the measure μ is uniform)

$\frac{a_1{b}_1+\cdots +a_n{b}_n}{n}\ge \frac{a_1+\cdots +a_n}{n}\frac{b_1+\cdots +b_n}{n}.$

More generally, for any probability measure μ on $\mathbb{ℝ}$ and increasing functions ƒ and g,

$\underset{\mathbb{ℝ}}{\int }f(x)g(x)d\mu (x)\ge \underset{\mathbb{ℝ}}{\int }f(x)d\mu (x)\underset{\mathbb{ℝ}}{\int }g(x)d\mu (x),$

which follows immediately from

$\underset{\mathbb{ℝ}}{\int }\underset{\mathbb{ℝ}}{\int }[f(x)-f(y)][g(x)-g(y)]d\mu (x)d\mu (y)\ge 0.$

The lattice condition is trivially satisfied also when the lattice is the product of totally ordered lattices, $X=X_1\times \cdots \times X_n$, and $\mu ={\mu }_1\otimes \cdots \otimes {\mu }_n$ is a product measure. Often all the factors (both the lattices and the measures) are identical, i.e., μ is the probability distribution of i.i.d. random variables.

The FKG inequality for the case of a product measure is known also as the Harris inequality after Harris Template:Harv, who found and used it in his study of percolation in the plane. A proof of the Harris inequality that uses the above double integral trick on $\mathbb{ℝ}$ can be found, e.g., in Section 2.2 of Template:Harvtxt.

Simple examples

A typical example is the following. Color each hexagon of the infinite honeycomb lattice black with probability $p$ and white with probability $1-p$, independently of each other. Let a, b, c, d be four hexagons, not necessarily distinct. Let $a\leftrightarrow b$ and $c\leftrightarrow d$ be the events that there is a black path from a to b, and a black path from c to d, respectively. Then the Harris inequality says that these events are positively correlated: $\mathrm{Pr}(a\leftrightarrow b,c\leftrightarrow d)\ge \mathrm{Pr}(a\leftrightarrow b)\mathrm{Pr}(c\leftrightarrow d)$. In other words, assuming the presence of one path can only increase the probability of the other.

Similarly, if we randomly color the hexagons inside an $n\times n$ rhombus-shaped hex board, then the events that there is black crossing from the left side of the board to the right side is positively correlated with having a black crossing from the top side to the bottom. On the other hand, having a left-to-right black crossing is negatively correlated with having a top-to-bottom white crossing, since the first is an increasing event (in the amount of blackness), while the second is decreasing. In fact, in any coloring of the hex board exactly one of these two events happen — this is why hex is a well-defined game.

In the Erdős–Rényi random graph, the existence of a Hamiltonian cycle is negatively correlated with the 3-colorability of the graph, since the first is an increasing event, while the latter is decreasing.

Beyond product measures

In statistical mechanics, the usual source of measures that satisfy the lattice condition (and hence the FKG inequality) is the following:

If $S$ is an ordered set (such as $\{-1,+1\}$), and $\mathrm{\Gamma }$ is a finite or infinite graph, then the set $S^{\mathrm{\Gamma }}$ of $S$-valued configurations is a poset that is a distributive lattice.

Now, if $\mathrm{\Phi }$ is a submodular potential (i.e., a family of functions

${\mathrm{\Phi }}_{\mathrm{\Lambda }}:S^{\mathrm{\Lambda }}⟶\mathbb{ℝ}\cup \{\mathrm{\infty }\},$

one for each finite $\mathrm{\Lambda }\subset \mathrm{\Gamma }$, such that each ${\mathrm{\Phi }}_{\mathrm{\Lambda }}$ is submodular), then one defines the corresponding Hamiltonians as

$H_{\mathrm{\Lambda }}(\varphi ):=\sum _{\mathrm{\Delta }\cap \mathrm{\Lambda }=\mathrm{\varnothing }}{\mathrm{\Phi }}_{\mathrm{\Delta }}(\varphi ).$

If μ is an extremal Gibbs measure for this Hamiltonian on the set of configurations $\varphi$, then it is easy to show that μ satisfies the lattice condition, see Template:Harvtxt.

A key example is the Ising model on a graph $\mathrm{\Gamma }$. Let $S=\{-1,+1\}$, called spins, and $\beta \in [0,\mathrm{\infty }]$. Take the following potential:

${\mathrm{\Phi }}_{\mathrm{\Lambda }}(\varphi )=\{\begin{array}{ll}\beta 1_{\{\varphi (x)=\varphi (y)\}}& \text{ if }\mathrm{\Lambda }=\{x,y\}\text{ is a pair of adjacent vertices of }\mathrm{\Gamma };\\ 0& \text{ otherwise.}\end{array}$

Submodularity is easy to check; intuitively, taking the min or the max of two configurations tends to decrease the number of disagreeing spins. Then, depending on the graph $\mathrm{\Gamma }$ and the value of $\beta$, there could be one or more extremal Gibbs measures, see, e.g., Template:Harvtxt and Template:Harvtxt.

A generalization: the Holley inequality

The Holley inequality, due to Template:Harvs, states that the expectations

$⟨f{⟩}_i=\frac{\sum _{x\in X}f(x){\mu }_i(x)}{\sum _{x\in X}{\mu }_i(x)}$

of a monotonically increasing function ƒ on a finite distributive lattice $X$ with respect to two positive functions μ₁, μ₂ on the lattice satisfy the condition

$⟨f{⟩}_1\ge ⟨f{⟩}_2,$

provided the functions satisfy the Holley condition (criterion)

${\mu }_2(x\wedge y){\mu }_1(x\vee y)\ge {\mu }_1(x){\mu }_2(y)$

for all x, y in the lattice.

To recover the FKG inequality: If μ satisfies the lattice condition and ƒ and g are increasing functions on $X$, then μ₁(x)=g(x)μ(x) and μ₂(x)=μ(x) will satisfy the lattice-type condition of the Holley inequality. Then the Holley inequality states that

$\frac{⟨fg{⟩}_{\mu }}{⟨g{⟩}_{\mu }}=⟨f{⟩}_1\ge ⟨f{⟩}_2=⟨f{⟩}_{\mu },$

which is just the FKG inequality.

As for FKG, the Holley inequality follows from the Ahlswede–Daykin inequality.

Weakening the lattice condition: monotonicity

Consider the usual case of $X$ being a product ${\mathbb{ℝ}}^V$ for some finite set $V$. The lattice condition on μ is easily seen to imply the following monotonicity, which has the virtue that it is often easier to check than the lattice condition:

Whenever one fixes a vertex $v\in V$ and two configurations φ and ψ outside v such that $\varphi (w)\ge \psi (w)$ for all $w=v$, the μ-conditional distribution of φ(v) given $\{\varphi (w):w=v\}$ stochastically dominates the μ-conditional distribution of ψ(v) given $\{\psi (w):w=v\}$.

Now, if μ satisfies this monotonicity property, that is already enough for the FKG inequality (positive associations) to hold.

Here is a rough sketch of the proof, due to Template:Harvtxt: starting from any initial configuration on $V$, one can run a simple Markov chain (the Metropolis algorithm) that uses independent Uniform[0,1] random variables to update the configuration in each step, such that the chain has a unique stationary measure, the given μ. The monotonicity of μ implies that the configuration at each step is a monotone function of independent variables, hence the product measure version of Harris implies that it has positive associations. Therefore, the limiting stationary measure μ also has this property.

The monotonicity property has a natural version for two measures, saying that μ₁ conditionally pointwise dominates μ₂. It is again easy to see that if μ₁ and μ₂ satisfy the lattice-type condition of the Holley inequality, then μ₁ conditionally pointwise dominates μ₂. On the other hand, a Markov chain coupling argument similar to the above, but now without invoking the Harris inequality, shows that conditional pointwise domination, in fact, implies stochastically domination. Stochastic domination is equivalent to saying that $⟨f{⟩}_1\ge ⟨f{⟩}_2$ for all increasing ƒ, thus we get a proof of the Holley inequality. (And thus also a proof of the FKG inequality, without using the Harris inequality.)

See Template:Harvtxt and Template:Harvtxt for details.

See also

• Ahlswede–Daykin inequality

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010