Height function: real map defined on the vertices of the tiling.
each tiling determined by its height function; the height is monotone w.r.t. flips.
Yields an intuitive lift inR3, used to prove connexity: Dimers (Thurston)
Rectangles (R´emila)
Kagome (Bodini)
T-tetraminoes(Korn-Pak)
Examples Questions Results Context-sensitive flips
Height function
Height function: real map defined on the vertices of the tiling.
Useful when
each tiling determined by its height function;
the height is monotone w.r.t. flips.
Yields an intuitive lift inR3, used to prove connexity: Dimers (Thurston)
Rectangles (R´emila)
Kagome (Bodini)
T-tetraminoes(Korn-Pak)
Height function
Height function: real map defined on the vertices of the tiling.
Useful when
each tiling determined by its height function;
the height is monotone w.r.t. flips.
Yields an intuitive lift inR3, used to prove connexity:
Dimers (Thurston)
Rectangles (R´emila)
Kagome (Bodini)
T-tetraminoes(Korn-Pak)
Examples Questions Results Context-sensitive flips
Diameter
Easy for dimers via height function.
Logarithmic in the number of tilings in any non-degenerate case.
Not necessarily a good indicator for mixing time. . .
Straightness
Straighttiling graph: any two tilings can be connected by a sequence of flips, with no flip being performed back and forth.
Holds for dimers (the height function orientates flips).
Holds for parallelogram iffn≤4 (Bodini-F.-Rao-R´emila 2011). Does it imply rapid mixing?
Examples Questions Results Context-sensitive flips
Limit shape
Well-studied for dimers(Kenyon, Okounkov, Propp. . . ). Other cases?
Limit shape
Well-studied for dimers(Kenyon, Okounkov, Propp. . . ). Other cases?
Examples Questions Results Context-sensitive flips
Limit shape
Well-studied for dimers(Kenyon, Okounkov, Propp. . . ). Other cases?
Examples Questions Results Context-sensitive flips
Mixing time
Rapid for dimers(Luby-Randall-Sinclair 1995, Randall-Tetali 1999, Wilson 2004, Caputo-Martinelli-Toninelli 2011, Laslier-Toninelli 2013).
Connexity still to be characterized in this case.
Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016). Claimed rapid for T-tetraminoes(Kayibi-Pirzada 2015).
Conjectured rapid for parallelograms whenn= 4 (Destainville 2001). Conjectured slow for parallelograms whenn= 4 (F. 2016).
Examples Questions Results Context-sensitive flips
Mixing time
Rapid for dimers(Luby-Randall-Sinclair 1995, Randall-Tetali 1999, Wilson 2004, Caputo-Martinelli-Toninelli 2011, Laslier-Toninelli 2013).
Rapid for Kagome without “fish”(Ugolnikova 2016). Connexity still to be characterized in this case.
Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016). Claimed rapid for T-tetraminoes(Kayibi-Pirzada 2015).
Conjectured rapid for parallelograms whenn= 4 (Destainville 2001). Conjectured slow for parallelograms whenn= 4 (F. 2016).
Examples Questions Results Context-sensitive flips
Mixing time
Rapid for dimers(Luby-Randall-Sinclair 1995, Randall-Tetali 1999, Wilson 2004, Caputo-Martinelli-Toninelli 2011, Laslier-Toninelli 2013).
Rapid for Kagome without “fish”(Ugolnikova 2016). Connexity still to be characterized in this case.
Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016).
Conjectured rapid for parallelograms whenn= 4 (Destainville 2001). Conjectured slow for parallelograms whenn= 4 (F. 2016).
Examples Questions Results Context-sensitive flips
Mixing time
Rapid for dimers(Luby-Randall-Sinclair 1995, Randall-Tetali 1999, Wilson 2004, Caputo-Martinelli-Toninelli 2011, Laslier-Toninelli 2013).
Rapid for Kagome without “fish”(Ugolnikova 2016). Connexity still to be characterized in this case.
Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016). Claimed rapid for T-tetraminoes(Kayibi-Pirzada 2015).
Conjectured rapid for parallelograms whenn= 4 (Destainville 2001). Conjectured slow for parallelograms whenn= 4 (F. 2016).
Examples Questions Results Context-sensitive flips
Mixing time
Rapid for dimers(Luby-Randall-Sinclair 1995, Randall-Tetali 1999, Wilson 2004, Caputo-Martinelli-Toninelli 2011, Laslier-Toninelli 2013).
Rapid for Kagome without “fish”(Ugolnikova 2016). Connexity still to be characterized in this case.
Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016). Claimed rapid for T-tetraminoes(Kayibi-Pirzada 2015).
Conjectured rapid for parallelograms whenn= 4 (Destainville 2001).
Examples Questions Results Context-sensitive flips
Mixing time
Rapid for dimers(Luby-Randall-Sinclair 1995, Randall-Tetali 1999, Wilson 2004, Caputo-Martinelli-Toninelli 2011, Laslier-Toninelli 2013).
Rapid for Kagome without “fish”(Ugolnikova 2016). Connexity still to be characterized in this case.
Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016). Claimed rapid for T-tetraminoes(Kayibi-Pirzada 2015).
Conjectured rapid for parallelograms whenn= 4 (Destainville 2001). Conjectured slow for parallelograms whenn= 4 (F. 2016).
1 Examples
2 Questions
3 Results
4 Context-sensitive flips
Examples Questions Results Context-sensitive flips
Principle
Consider the Markov chain
choose a vertex of the tiling uniformly at random,
perform a flip around it with probability P = exp(−∆E)/T, where
T is a temperatureparameter,
theenergy E of a tiling is a sum of local interactions.
This is the Metropolis-Hastings algorithm for the Gibbs distribution P(tiling) = 1
It aims to model the cooling of ordered materials (quasicrystals).
Principle
Consider the Markov chain
choose a vertex of the tiling uniformly at random,
perform a flip around it with probability P = exp(−∆E)/T, where
T is a temperatureparameter,
theenergy E of a tiling is a sum of local interactions.
This is the Metropolis-Hastings algorithm for the Gibbs distribution P(tiling) = 1
It aims to model the cooling of ordered materials (quasicrystals).
Examples Questions Results Context-sensitive flips
Example 1: dimer tilings
Ergodic atT >0, Θ(n2) mixing at T =∞,O(n2.5) atT = 0.
Example 1: dimer tilings
Ergodic atT >0, Θ(n2) mixing at T =∞,Θ(n2) atT = 0.
Examples Questions Results Context-sensitive flips
Example 1: dimer tilings
Ergodic atT >0, Θ(n2) mixing at T =∞,Θ(n2) atT = 0.
Example 1: dimer tilings
Ergodic atT >0, Θ(n2) mixing at T =∞,Θ(n2) atT = 0.
Examples Questions Results Context-sensitive flips
Example 1: dimer tilings
Ergodic atT >0, Θ(n2) mixing at T =∞,Θ(n2) atT = 0.
Example 1: dimer tilings
Ergodic atT >0, Θ(n2) mixing at T =∞,Θ(n2) atT = 0.
Examples Questions Results Context-sensitive flips
Example 2: Beenker tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Example 2: Beenker tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Examples Questions Results Context-sensitive flips
Example 2: Beenker tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Example 2: Beenker tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Examples Questions Results Context-sensitive flips
Example 2: Beenker tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Example 3: Penrose tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Examples Questions Results Context-sensitive flips
Example 3: Penrose tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Example 3: Penrose tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Examples Questions Results Context-sensitive flips
Example 3: Penrose tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.
Example 3: Penrose tilings
Ergodic atT >0,unknown mixing atT =∞,Θ(n2)at T = 0.