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The ratio between the number of nearly zero energy states induced by the magnetic field and the number of hexagons, N$_{D}$ ( φ ) N$_{h}$ , should depend on a dimensionless quantity consisting of a combination of ℓ , a$_{0}$ , L$_{x}$ and L$_{y}$ , which are the important parameters of rectangu- lar graphene dots. The lengths L$_{x}$ and L$_{y}$ should ap- pear as L$_{x}$ + L$_{y}$ so that for rectangular-shaped graphene sheets N$_{D}$/N$_{h}$ remains the same when L$_{x}$ and L$_{y}$ are ex- changed. These considerations lead us to the dimension- less variable ℓ 2 a 0 ( L$_{x}$ + L$_{y}$ ) = 3 4 π 1 φ ( M $_{+}$√$_{3}$$_{N}$$_{+2}$$_{/}$ √$_{3)}$ , see Eq.(3). We are especially interested in the regime where the di- ameter of the cyclotron orbit is comparable to the sys- 

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tem length 2 ℓ ∼ L$_{x,y}$ . Note that in this regime many cyclotron orbits get affected by the presence of the edges and corners of the rectangular dot. Since we must also assume that Hofstadter effect is negligible the validity regime of Eq.(3) is a$_{0}$ ≪ 2 ℓ < L$_{x,y}$ . Note also that the scaling function f ( x ) should be different for each δ . 

We first show how the wavefunction of a non-zero en- ergy state at φ = 0 changes into a state with nearly zero energy as φ increases. Consider the probability wave- function for n = 1027 at a finite φ = 0 . 01, as shown in 

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## III. WAVEFUNCTIONS OF QUASI-DEGENERATE STATES IN A MAGNETIC FIELD 

