**Information theory has been closely related to concepts of statistical mechanics, almost from its inception. In recent years this connection has been extended to encompass: error correcting codes based on random graphs on one hand, and spin glasses on the other hand. In a nutshell, these systems display phase transitions of similar nature.**\\
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**Phase transitions** are an ubiquitous natural phenomenon and occur in almost all physical situations involving a macroscopic number of interacting degrees of freedom. The most well known ones are displayed in the **phase diagram of ordinary matter** (left picture). The system undergoes sudden transitions between liquid and solid or liquid and vapour phases when the control parameters such as pressure and/or temperature cross the solid lines. Simple but very important

|{{en:research:statmech:phasediagram2.jpg?250x200|phase diagram of water}}| | | |{{en:research:statmech:squarelattice.jpg?250x200|simplest Ising model}}|

models which capture essential features of phase transitions belong to the class of **Ising spin systems** (right picture). Binary variables (also called spins) taking values ''+1/-1'', and representing the presence/absence of a water molecule are attached to the red circles of the lattice. The blue squares indicate that there is an energy cost depending on the pair ''(+1,+1)'' ''(+1,-1)'' ''(-1,+1)'' ''(-1,-1)'' joined by the edge. Each configuration of spins on the lattice is assigned a probability weight depending on the total energy cost. The main qualitative features of the water-vapour phase transition are captured by the typical configurations of the **Gibbs probability weight**. 

**Modern error correcting codes** are also based on probabilistic graphical models.
For example, in **Low Density Parity Check Codes** (''LDPC'') the code words are  strings of bits (''1 and 0'') attached to the red nodes, satisfying a set of linear constraints depicted by the graph below. All the bits that are connected to a square blue node (a parity check) sum to zero modulo ''2''.
The code words are sent through a communication channel which flips each bit with a certain probability ''p'' (the noise level). As such, **the code bits can be viewed as set of interacting spins and the communication system can be modelled by a probabilistic graphical model belonging to the Ising class**. The system displays a phase transition between a phase ''p<p_c'' where it 

|{{en:research:statmech:ldpc36.jpg?250x120|LDPC(3,6)}}| | | {{en:research:statmech:phasediagramcode.jpg?250x120|phase diagram of LDPC}}|

is in principle possible to clean the errors and reconstruct perfectly the original code word; and a phase ''p>p_c'' where this is impossible. However for ''p<p_c'' it is not always possible to perform the error correction by an efficient algorithm. It turns out that a fundamental **efficient algorithm**, called **belief propagation** and related to **mean field methods of statistical physics**, can  efficiently correct the errors for ''p<p_d''. These phase transitions share deep similarities with the ones occuring in **spin glasses** which are spin systems based on **random Gibbs weights**. Indeed, the probabilistic models for graphical coding schemes have quenched randomness coming from the underlying graph code as well as the channel output realizations.