perature dependence of the spectroscopic gap ( Δ( r, T ) ) is inhomogenous, and that the gap vanishes at a local temperature T$_{p}$ ( r ) , which is positively correlated in space with the gap size[1]. Furthermore, the spectroscopic gap be- low T$_{c}$ is spatially correlated to the density of states at the Fermi level well above T$_{c}$ where the gap is absent[12]. For this work we simplify the analysis of a large number of points by us- ing a simple, operational definition for the gap and the pairing temperature. We follow the def- inition used previously[1, 2], defining the gap value as the point of maximum differential con- ductance for positive bias. This simple proce- dure agrees with more elaborate fitting schemes to within 10% in most cases. We also need a simple method to describe the temperature evo- lution of the gap. For this, we define the pairing temperature T$_{p}$ , which is the highest tempera- ture at which a maximum in the dI/dV spec- trum can be observed within experimental reso- lution at positive bias. 

In underdoped samples, multiple energy features are present in STM spectroscopy[1]. While high energy features survive to temper- atures well above T$_{c}$ , lower energy features show the closing of a nodal gap near T$_{c}$ [13, 14]. However, for strongly overdoped sam- ples pseudogap behavior is absent, there is only one spectroscopic feature in tunneling data, and the median value of T$_{p}$ is near the resis- tive transition temperature T$_{c}$ . Furthermore, 

an inhomogeneous superconducting transition is supported by Nernst measurements[15] and µ SR[16]. Therefore, Δ may be taken as the su- perconducting gap, and T$_{p}$ as the temperature for the onset of pairing. We have done detailed measurements of the local T$_{p}$ and the local Δ for a grid of points on an overdoped sample of Bi-2212 with T$_{c}$ = 65 K, for a range of temper- atures from 50 K to 76 K. These measurements provided evidence that nanoscale regions distin- guished by inhomogeneity can influence each other via the proximity effect. We show that the fractional variation in T$_{p}$ is less than that for Δ . The pairing order parameter in a disordered sample has been analyzed theoretically at low temperature[17, 18] and near the transition[19], but a direct comparison of spatial variation be- tween T$_{p}$ and Δ was not made. Our measure- ments show that the length scale for correlations in T$_{p}$ is longer by 20% than the length scale for the gap. Furthermore, we find that the local gap can disappear at distinct temperatures for dif- ferent spatial positions on the sample, even if these positions have the same magnitude of gap at low temperature, and this anomalous temper- ature evolution of the local gap is influenced by the size of gaps in the surrounding 1 nm region. 

We have the spectral evolution for more than 10,000 points in a 25 x 28 nm grid[21] at four- teen different temperatures from 50 K to 76 K. For each temperature, we can determine the gap at each point, and plot a series of “gap maps”, 

