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Critical resonance in the non-intersecting lattice path model. Richard W. Kenyon Université Paris-Sud. David B. Wilson Microsoft Research. Non-intersecting lattice paths. Partition Function Z. Weights a,b,c and the Phases. a. Solid `a’ type edges. Solid `b’ type edges.

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Critical resonance in the non-intersecting lattice path model


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critical resonance in the non intersecting lattice path model

Critical resonance in the non-intersecting lattice path model

Richard W. Kenyon

Université Paris-Sud

David B. Wilson

Microsoft Research

weights a b c and the phases
Weights a,b,c and the Phases

a

Solid `a’ type edges

Solid `b’ type edges

Solid `c’ type edges

c

b

kasteleyn matrices
If m, n even, each loop has even length, (-1)=(-1)#loopsKasteleyn Matrices

Topology: If a loop windsptimes horizontally,qtimes vertically,

then all loops wind (p,q) times, where gcd(p,q)=1

Weight horizontal cut-edges by (-1)

vertical cut-edges by (-1)

ratcheting
Ratcheting(?)

vs

Many loops, each winding around once

One loop, winding around many times

free fermions in 1 dimension
Free fermions in 1 dimension

Fermions are particles that repel one another.

In 1 dimension, log(Z)/ -Li3/2(-)

density / -Li1/2(-).

Same equations for lattice paths when (presumed) ratchet number is 0.

When (presumed) ratchet number is nonzero,

behavior differs from free fermions.

resonance
Resonance

Many physical systems exhibit resonance when ratios of frequencies nearly rational.

Does resonance for lattice paths resemble resonance in other systems?